Calculus Examples

$\frac{d y}{d x} - \frac{y^{2} - 1}{x} = 0$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Solve for $\frac{d y}{d x}$ .

Tap for more steps...

Step 1.1.1

Simplify $\frac{d y}{d x} - \frac{y^{2} - 1}{x}$ .

Tap for more steps...

Step 1.1.1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{d y}{d x} - \frac{y^{2} - 1^{2}}{x} = 0$

Step 1.1.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\frac{d y}{d x} - \frac{(y + 1) (y - 1)}{x} = 0$

Step 1.1.1.2

To write $\frac{d y}{d x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{\frac{d y}{d x} x}{x} - \frac{(y + 1) (y - 1)}{x} = 0$

Step 1.1.1.3

Combine the numerators over the common denominator.

$\frac{\frac{d y}{d x} x - (y + 1) (y - 1)}{x} = 0$

Step 1.1.1.4

Simplify the numerator.

Tap for more steps...

Step 1.1.1.4.1

Apply the distributive property.

$\frac{\frac{d y}{d x} x + (- y - 1 \cdot 1) (y - 1)}{x} = 0$

Step 1.1.1.4.2

Multiply $- 1$ by $1$ .

$\frac{\frac{d y}{d x} x + (- y - 1) (y - 1)}{x} = 0$

Step 1.1.1.4.3

Expand $(- y - 1) (y - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.1.1.4.3.1

Apply the distributive property.

$\frac{\frac{d y}{d x} x - y (y - 1) - 1 (y - 1)}{x} = 0$

Step 1.1.1.4.3.2

Apply the distributive property.

$\frac{\frac{d y}{d x} x - y \cdot y - y \cdot - 1 - 1 (y - 1)}{x} = 0$

Step 1.1.1.4.3.3

Apply the distributive property.

$\frac{\frac{d y}{d x} x - y \cdot y - y \cdot - 1 - 1 y - 1 \cdot - 1}{x} = 0$

Step 1.1.1.4.4

Simplify and combine like terms.

Tap for more steps...

Step 1.1.1.4.4.1

Simplify each term.

Tap for more steps...

Step 1.1.1.4.4.1.1

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 1.1.1.4.4.1.1.1

Move $y$ .

$\frac{\frac{d y}{d x} x - (y \cdot y) - y \cdot - 1 - 1 y - 1 \cdot - 1}{x} = 0$

Step 1.1.1.4.4.1.1.2

Multiply $y$ by $y$ .

$\frac{\frac{d y}{d x} x - y^{2} - y \cdot - 1 - 1 y - 1 \cdot - 1}{x} = 0$

Step 1.1.1.4.4.1.2

Multiply $- y \cdot - 1$ .

Tap for more steps...

Step 1.1.1.4.4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{d y}{d x} x - y^{2} + 1 y - 1 y - 1 \cdot - 1}{x} = 0$

Step 1.1.1.4.4.1.2.2

Multiply $y$ by $1$ .

$\frac{\frac{d y}{d x} x - y^{2} + y - 1 y - 1 \cdot - 1}{x} = 0$

Step 1.1.1.4.4.1.3

Rewrite $- 1 y$ as $- y$ .

$\frac{\frac{d y}{d x} x - y^{2} + y - y - 1 \cdot - 1}{x} = 0$

Step 1.1.1.4.4.1.4

Multiply $- 1$ by $- 1$ .

$\frac{\frac{d y}{d x} x - y^{2} + y - y + 1}{x} = 0$

Step 1.1.1.4.4.2

Subtract $y$ from $y$ .

$\frac{\frac{d y}{d x} x - y^{2} + 0 + 1}{x} = 0$

Step 1.1.1.4.4.3

Add $- y^{2}$ and $0$ .

$\frac{\frac{d y}{d x} x - y^{2} + 1}{x} = 0$

Step 1.1.2

Set the numerator equal to zero.

$\frac{d y}{d x} x - y^{2} + 1 = 0$

Step 1.1.3

Solve the equation for $\frac{d y}{d x}$ .

Tap for more steps...

Step 1.1.3.1

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

Tap for more steps...

Step 1.1.3.1.1

Add $y^{2}$ to both sides of the equation.

$\frac{d y}{d x} x + 1 = y^{2}$

Step 1.1.3.1.2

Subtract $1$ from both sides of the equation.

$\frac{d y}{d x} x = y^{2} - 1$

Step 1.1.3.2

Divide each term in $\frac{d y}{d x} x = y^{2} - 1$ by $x$ and simplify.

Tap for more steps...

Step 1.1.3.2.1

Divide each term in $\frac{d y}{d x} x = y^{2} - 1$ by $x$ .

$\frac{\frac{d y}{d x} x}{x} = \frac{y^{2}}{x} + \frac{- 1}{x}$

Step 1.1.3.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.3.2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.1.3.2.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} x}{x} = \frac{y^{2}}{x} + \frac{- 1}{x}$

Step 1.1.3.2.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{y^{2}}{x} + \frac{- 1}{x}$

Step 1.1.3.2.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.2.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = \frac{y^{2}}{x} - \frac{1}{x}$

Step 1.2

Factor.

Tap for more steps...

Step 1.2.1

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{y^{2} - 1}{x}$

Step 1.2.2

Simplify the numerator.

Tap for more steps...

Step 1.2.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{d y}{d x} = \frac{y^{2} - 1^{2}}{x}$

Step 1.2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\frac{d y}{d x} = \frac{(y + 1) (y - 1)}{x}$

Step 1.3

Multiply both sides by $\frac{1}{(y + 1) (y - 1)}$ .

$\frac{1}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{1}{(y + 1) (y - 1)} \cdot \frac{(y + 1) (y - 1)}{x}$

Step 1.4

Cancel the common factor of $(y + 1) (y - 1)$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$\frac{1}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{1}{(y + 1) (y - 1)} \cdot \frac{(y + 1) (y - 1)}{x}$

Step 1.4.2

Rewrite the expression.

$\frac{1}{(y + 1) (y - 1)} \frac{d y}{d x} = \frac{1}{x}$

Step 1.5

Rewrite the equation.

$\frac{1}{(y + 1) (y - 1)} d y = \frac{1}{x} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{(y + 1) (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Write the fraction using partial fraction decomposition.

Tap for more steps...

Step 2.2.1.1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 2.2.1.1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{y + 1}$

Step 2.2.1.1.2

$\frac{A}{y + 1} + \frac{B}{y - 1}$

Step 2.2.1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(y + 1) (y - 1)$ .

$\frac{1 (y + 1) (y - 1)}{(y + 1) (y - 1)} = \frac{(A) (y + 1) (y - 1)}{y + 1} + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.4

Cancel the common factor of $y + 1$ .

Tap for more steps...

Step 2.2.1.1.4.1

Cancel the common factor.

$\frac{1 (y + 1) (y - 1)}{(y + 1) (y - 1)} = \frac{(A) (y + 1) (y - 1)}{y + 1} + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.4.2

Rewrite the expression.

$\frac{1 (y - 1)}{y - 1} = \frac{(A) (y + 1) (y - 1)}{y + 1} + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.5

Cancel the common factor of $y - 1$ .

Tap for more steps...

Step 2.2.1.1.5.1

Cancel the common factor.

$\frac{1 (y - 1)}{y - 1} = \frac{(A) (y + 1) (y - 1)}{y + 1} + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.5.2

Rewrite the expression.

$1 = \frac{(A) (y + 1) (y - 1)}{y + 1} + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.6

Simplify each term.

Tap for more steps...

Step 2.2.1.1.6.1

Cancel the common factor of $y + 1$ .

Tap for more steps...

Step 2.2.1.1.6.1.1

Cancel the common factor.

$1 = \frac{A (y + 1) (y - 1)}{y + 1} + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.6.1.2

Divide $(A) (y - 1)$ by $1$ .

$1 = (A) (y - 1) + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.6.2

Apply the distributive property.

$1 = A y + A \cdot - 1 + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.6.3

Move $- 1$ to the left of $A$ .

$1 = A y - 1 \cdot A + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.6.4

Rewrite $- 1 A$ as $- A$ .

$1 = A y - A + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.6.5

Cancel the common factor of $y - 1$ .

Tap for more steps...

Step 2.2.1.1.6.5.1

Cancel the common factor.

$1 = A y - A + \frac{(B) (y + 1) (y - 1)}{y - 1}$

Step 2.2.1.1.6.5.2

Divide $(B) (y + 1)$ by $1$ .

$1 = A y - A + (B) (y + 1)$

Step 2.2.1.1.6.6

Apply the distributive property.

$1 = A y - A + B y + B \cdot 1$

Step 2.2.1.1.6.7

Multiply $B$ by $1$ .

$1 = A y - A + B y + B$

Step 2.2.1.1.7

Move $- A$ .

$1 = A y + B y - A + B$

Step 2.2.1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

Tap for more steps...

Step 2.2.1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $y$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + B$

Step 2.2.1.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $y$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = - 1 A + B$

Step 2.2.1.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = A + B$

$1 = - 1 A + B$

$0 = A + B$

$1 = - 1 A + B$

Step 2.2.1.3

Solve the system of equations.

Tap for more steps...

Step 2.2.1.3.1

Solve for $A$ in $0 = A + B$ .

Tap for more steps...

Step 2.2.1.3.1.1

Rewrite the equation as $A + B = 0$ .

$A + B = 0$

$1 = - 1 A + B$

Step 2.2.1.3.1.2

Subtract $B$ from both sides of the equation.

$A = - B$

$1 = - 1 A + B$

$A = - B$

$1 = - 1 A + B$

Step 2.2.1.3.2

Replace all occurrences of $A$ with $- B$ in each equation.

Tap for more steps...

Step 2.2.1.3.2.1

Replace all occurrences of $A$ in $1 = - 1 A + B$ with $- B$ .

$1 = - 1 (- B) + B$

$A = - B$

Step 2.2.1.3.2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1.3.2.2.1

Simplify $- 1 (- B) + B$ .

Tap for more steps...

Step 2.2.1.3.2.2.1.1

Multiply $- 1 (- B)$ .

Tap for more steps...

Step 2.2.1.3.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$1 = 1 B + B$

$A = - B$

Step 2.2.1.3.2.2.1.1.2

Multiply $B$ by $1$ .

$1 = B + B$

$A = - B$

$1 = B + B$

$A = - B$

Step 2.2.1.3.2.2.1.2

Add $B$ and $B$ .

$1 = 2 B$

$A = - B$

$1 = 2 B$

$A = - B$

$1 = 2 B$

$A = - B$

$1 = 2 B$

$A = - B$

Step 2.2.1.3.3

Solve for $B$ in $1 = 2 B$ .

Tap for more steps...

Step 2.2.1.3.3.1

Rewrite the equation as $2 B = 1$ .

$2 B = 1$

$A = - B$

Step 2.2.1.3.3.2

Divide each term in $2 B = 1$ by $2$ and simplify.

Tap for more steps...

Step 2.2.1.3.3.2.1

Divide each term in $2 B = 1$ by $2$ .

$\frac{2 B}{2} = \frac{1}{2}$

$A = - B$

Step 2.2.1.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1.3.3.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.3.3.2.2.1.1

Cancel the common factor.

$\frac{2 B}{2} = \frac{1}{2}$

$A = - B$

Step 2.2.1.3.3.2.2.1.2

Divide $B$ by $1$ .

$B = \frac{1}{2}$

$A = - B$

$B = \frac{1}{2}$

$A = - B$

$B = \frac{1}{2}$

$A = - B$

$B = \frac{1}{2}$

$A = - B$

$B = \frac{1}{2}$

$A = - B$

Step 2.2.1.3.4

Replace all occurrences of $B$ with $\frac{1}{2}$ in each equation.

Tap for more steps...

Step 2.2.1.3.4.1

Replace all occurrences of $B$ in $A = - B$ with $\frac{1}{2}$ .

$A = - (\frac{1}{2})$

$B = \frac{1}{2}$

Step 2.2.1.3.4.2

Simplify the right side.

Tap for more steps...

Step 2.2.1.3.4.2.1

Multiply $- 1$ by $\frac{1}{2}$ .

$A = - \frac{1}{2}$

$B = \frac{1}{2}$

$A = - \frac{1}{2}$

$B = \frac{1}{2}$

$A = - \frac{1}{2}$

$B = \frac{1}{2}$

Step 2.2.1.3.5

List all of the solutions.

$A = - \frac{1}{2}, B = \frac{1}{2}$

Step 2.2.1.4

Replace each of the partial fraction coefficients in $\frac{A}{y + 1} + \frac{B}{y - 1}$ with the values found for $A$ and $B$ .

$\frac{- \frac{1}{2}}{y + 1} + \frac{\frac{1}{2}}{y - 1}$

Step 2.2.1.5

Simplify.

Tap for more steps...

Step 2.2.1.5.1

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{2} \cdot \frac{1}{y + 1} + \frac{\frac{1}{2}}{y - 1}$

Step 2.2.1.5.2

Multiply $\frac{1}{y + 1}$ by $\frac{1}{2}$ .

$- \frac{1}{(y + 1) \cdot 2} + \frac{\frac{1}{2}}{y - 1}$

Step 2.2.1.5.3

Move $2$ to the left of $y + 1$ .

$- \frac{1}{2 (y + 1)} + \frac{\frac{1}{2}}{y - 1}$

Step 2.2.1.5.4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{2 (y + 1)} + \frac{1}{2} \cdot \frac{1}{y - 1}$

Step 2.2.1.5.5

Multiply $\frac{1}{2}$ by $\frac{1}{y - 1}$ .

$\int - \frac{1}{2 (y + 1)} + \frac{1}{2 (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2.2

Split the single integral into multiple integrals.

$\int - \frac{1}{2 (y + 1)} d y + \int \frac{1}{2 (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2.3

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$- \int \frac{1}{2 (y + 1)} d y + \int \frac{1}{2 (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2.4

Since $\frac{1}{2}$ is constant with respect to $y$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int \frac{1}{y + 1} d y) + \int \frac{1}{2 (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2.5

Let $u_{1} = y + 1$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.2.5.1

Let $u_{1} = y + 1$ . Find $\frac{d u_{1}}{d y}$ .

Tap for more steps...

Step 2.2.5.1.1

Differentiate $y + 1$ .

$\frac{d}{d y} [y + 1]$

Step 2.2.5.1.2

By the Sum Rule, the derivative of $y + 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [1]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d y} [1]$

Step 2.2.5.1.4

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$1 + 0$

Step 2.2.5.1.5

Add $1$ and $0$ .

$1$

Step 2.2.5.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$- \frac{1}{2} \int \frac{1}{u_{1}} d u_{1} + \int \frac{1}{2 (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2.6

The integral of $\frac{1}{u_{1}}$ with respect to $u_{1}$ is $\ln (| u_{1} |)$ .

$- \frac{1}{2} (\ln (| u_{1} |) + C_{1}) + \int \frac{1}{2 (y - 1)} d y = \int \frac{1}{x} d x$

Step 2.2.7

Since $\frac{1}{2}$ is constant with respect to $y$ , move $\frac{1}{2}$ out of the integral.

$- \frac{1}{2} (\ln (| u_{1} |) + C_{1}) + \frac{1}{2} \int \frac{1}{y - 1} d y = \int \frac{1}{x} d x$

Step 2.2.8

Let $u_{2} = y - 1$ . Then $d u_{2} = d y$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 2.2.8.1

Let $u_{2} = y - 1$ . Find $\frac{d u_{2}}{d y}$ .

Tap for more steps...

Step 2.2.8.1.1

Differentiate $y - 1$ .

$\frac{d}{d y} [y - 1]$

Step 2.2.8.1.2

By the Sum Rule, the derivative of $y - 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$

Step 2.2.8.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d y} [- 1]$

Step 2.2.8.1.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- 1$ with respect to $y$ is $0$ .

$1 + 0$

Step 2.2.8.1.5

Add $1$ and $0$ .

$1$

Step 2.2.8.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$- \frac{1}{2} (\ln (| u_{1} |) + C_{1}) + \frac{1}{2} \int \frac{1}{u_{2}} d u_{2} = \int \frac{1}{x} d x$

Step 2.2.9

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$- \frac{1}{2} (\ln (| u_{1} |) + C_{1}) + \frac{1}{2} (\ln (| u_{2} |) + C_{2}) = \int \frac{1}{x} d x$

Step 2.2.10

Simplify.

$- \frac{1}{2} \ln (| u_{1} |) + \frac{1}{2} \ln (| u_{2} |) + C_{3} = \int \frac{1}{x} d x$

Step 2.2.11

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 2.2.11.1

Replace all occurrences of $u_{1}$ with $y + 1$ .

$- \frac{1}{2} \ln (| y + 1 |) + \frac{1}{2} \ln (| u_{2} |) + C_{3} = \int \frac{1}{x} d x$

Step 2.2.11.2

Replace all occurrences of $u_{2}$ with $y - 1$ .

$- \frac{1}{2} \ln (| y + 1 |) + \frac{1}{2} \ln (| y - 1 |) + C_{3} = \int \frac{1}{x} d x$

Step 2.3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$- \frac{1}{2} \ln (| y + 1 |) + \frac{1}{2} \ln (| y - 1 |) + C_{3} = \ln (| x |) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$- \frac{1}{2} \ln (| y + 1 |) + \frac{1}{2} \ln (| y - 1 |) = \ln (| x |) + C$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Combine $\ln (| y + 1 |)$ and $\frac{1}{2}$ .

$- \frac{\ln (| y + 1 |)}{2} + \frac{1}{2} \ln (| y - 1 |) = \ln (| x |) + C$

Step 3.1.1.2

Combine $\frac{1}{2}$ and $\ln (| y - 1 |)$ .

$- \frac{\ln (| y + 1 |)}{2} + \frac{\ln (| y - 1 |)}{2} = \ln (| x |) + C$

Step 3.2

Multiply each term in $- \frac{\ln (| y + 1 |)}{2} + \frac{\ln (| y - 1 |)}{2} = \ln (| x |) + C$ by $2$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $- \frac{\ln (| y + 1 |)}{2} + \frac{\ln (| y - 1 |)}{2} = \ln (| x |) + C$ by $2$ .

$- \frac{\ln (| y + 1 |)}{2} \cdot 2 + \frac{\ln (| y - 1 |)}{2} \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.1.1

Move the leading negative in $- \frac{\ln (| y + 1 |)}{2}$ into the numerator.

$\frac{- \ln (| y + 1 |)}{2} \cdot 2 + \frac{\ln (| y - 1 |)}{2} \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 3.2.2.1.1.2

Cancel the common factor.

$\frac{- \ln (| y + 1 |)}{2} \cdot 2 + \frac{\ln (| y - 1 |)}{2} \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 3.2.2.1.1.3

Rewrite the expression.

$- \ln (| y + 1 |) + \frac{\ln (| y - 1 |)}{2} \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 3.2.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.2.1

Cancel the common factor.

$- \ln (| y + 1 |) + \frac{\ln (| y - 1 |)}{2} \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 3.2.2.1.2.2

Rewrite the expression.

$- \ln (| y + 1 |) + \ln (| y - 1 |) = \ln (| x |) \cdot 2 + C \cdot 2$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Move $2$ to the left of $\ln (| x |)$ .

$- \ln (| y + 1 |) + \ln (| y - 1 |) = 2 \cdot \ln (| x |) + C \cdot 2$

Step 3.2.3.1.2

Move $2$ to the left of $C$ .

$- \ln (| y + 1 |) + \ln (| y - 1 |) = 2 \ln (| x |) + 2 C$

Step 3.3

Move all the terms containing a logarithm to the left side of the equation.

$- \ln (| y + 1 |) + \ln (| y - 1 |) - 2 \ln (| x |) = 2 C$

Step 3.4

Simplify the left side.

Tap for more steps...

Step 3.4.1

Simplify $- \ln (| y + 1 |) + \ln (| y - 1 |) - 2 \ln (| x |)$ .

Tap for more steps...

Step 3.4.1.1

Simplify each term.

Tap for more steps...

Step 3.4.1.1.1

Simplify $- 2 \ln (| x |)$ by moving $2$ inside the logarithm.

$- \ln (| y + 1 |) + \ln (| y - 1 |) - \ln ({| x |}^{2}) = 2 C$

Step 3.4.1.1.2

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$- \ln (| y + 1 |) + \ln (| y - 1 |) - \ln (x^{2}) = 2 C$

Step 3.4.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- \ln (| y + 1 |) + \ln (\frac{| y - 1 |}{x^{2}}) = 2 C$

Step 3.5

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{| y - 1 |}{x^{2}})} = e^{2 C + \ln (| y + 1 |)}$

Step 3.6

Rewrite $\ln (\frac{| y - 1 |}{x^{2}}) = 2 C + \ln (| y + 1 |)$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{2 C + \ln (| y + 1 |)} = \frac{| y - 1 |}{x^{2}}$

Step 3.7

Solve for $y$ .

Tap for more steps...

Step 3.7.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 C + \ln (| y + 1 |)}) = \ln (\frac{| y - 1 |}{x^{2}})$

Step 3.7.2

Expand the left side.

Tap for more steps...

Step 3.7.2.1

Expand $\ln (e^{2 C + \ln (| y + 1 |)})$ by moving $2 C + \ln (| y + 1 |)$ outside the logarithm.

$(2 C + \ln (| y + 1 |)) \ln (e) = \ln (\frac{| y - 1 |}{x^{2}})$

Step 3.7.2.2

The natural logarithm of $e$ is $1$ .

$(2 C + \ln (| y + 1 |)) \cdot 1 = \ln (\frac{| y - 1 |}{x^{2}})$

Step 3.7.2.3

Multiply $2 C + \ln (| y + 1 |)$ by $1$ .

$2 C + \ln (| y + 1 |) = \ln (\frac{| y - 1 |}{x^{2}})$

Step 3.7.3

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| y + 1 |) - \ln (\frac{| y - 1 |}{x^{2}}) = - 2 C$

Step 3.7.4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| y + 1 |}{\frac{| y - 1 |}{x^{2}}}) = - 2 C$

Step 3.7.5

Multiply the numerator by the reciprocal of the denominator.

$\ln (| y + 1 | (\frac{x^{2}}{| y - 1 |})) = - 2 C$

Step 3.7.6

Combine $| y + 1 |$ and $\frac{x^{2}}{| y - 1 |}$ .

$\ln (\frac{| y + 1 | x^{2}}{| y - 1 |}) = - 2 C$

Step 3.7.7

Reorder factors in $\ln (\frac{| y + 1 | x^{2}}{| y - 1 |})$ .

$\ln (\frac{x^{2} | y + 1 |}{| y - 1 |}) = - 2 C$

Step 3.7.8

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{x^{2} | y + 1 |}{| y - 1 |})} = e^{- 2 C}$

Step 3.7.9

Rewrite $\ln (\frac{x^{2} | y + 1 |}{| y - 1 |}) = - 2 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 2 C} = \frac{x^{2} | y + 1 |}{| y - 1 |}$

Step 3.7.10

Solve for $y$ .

Tap for more steps...

Step 3.7.10.1

Rewrite the equation as $\frac{x^{2} | y + 1 |}{| y - 1 |} = e^{- 2 C}$ .

$\frac{x^{2} | y + 1 |}{| y - 1 |} = e^{- 2 C}$

Step 3.7.10.2

Multiply both sides by $| y - 1 |$ .

$\frac{x^{2} | y + 1 |}{| y - 1 |} | y - 1 | = e^{- 2 C} | y - 1 |$

Step 3.7.10.3

Simplify the left side.

Tap for more steps...

Step 3.7.10.3.1

Cancel the common factor of $| y - 1 |$ .

Tap for more steps...

Step 3.7.10.3.1.1

Cancel the common factor.

$\frac{x^{2} | y + 1 |}{| y - 1 |} | y - 1 | = e^{- 2 C} | y - 1 |$

Step 3.7.10.3.1.2

Rewrite the expression.

$x^{2} | y + 1 | = e^{- 2 C} | y - 1 |$

Step 3.7.10.4

Solve for $y$ .

Tap for more steps...

Step 3.7.10.4.1

Rewrite the equation as $e^{- 2 C} | y - 1 | = x^{2} | y + 1 |$ .

$e^{- 2 C} | y - 1 | = x^{2} | y + 1 |$

Step 3.7.10.4.2

Divide each term in $e^{- 2 C} | y - 1 | = x^{2} | y + 1 |$ by $e^{- 2 C}$ and simplify.

Tap for more steps...

Step 3.7.10.4.2.1

Divide each term in $e^{- 2 C} | y - 1 | = x^{2} | y + 1 |$ by $e^{- 2 C}$ .

$\frac{e^{- 2 C} | y - 1 |}{e^{- 2 C}} = \frac{x^{2} | y + 1 |}{e^{- 2 C}}$

Step 3.7.10.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.7.10.4.2.2.1

Cancel the common factor of $e^{- 2 C}$ .

Tap for more steps...

Step 3.7.10.4.2.2.1.1

Cancel the common factor.

$\frac{e^{- 2 C} | y - 1 |}{e^{- 2 C}} = \frac{x^{2} | y + 1 |}{e^{- 2 C}}$

Step 3.7.10.4.2.2.1.2

Divide $| y - 1 |$ by $1$ .

$| y - 1 | = \frac{x^{2} | y + 1 |}{e^{- 2 C}}$

Step 3.7.10.4.3

Rewrite the absolute value equation as four equations without absolute value bars.

$y - 1 = \frac{x^{2} (y + 1)}{e^{- 2 C}}$

$y - 1 = - \frac{x^{2} (y + 1)}{e^{- 2 C}}$

$- (y - 1) = \frac{x^{2} (y + 1)}{e^{- 2 C}}$

$- (y - 1) = - \frac{x^{2} (y + 1)}{e^{- 2 C}}$

Step 3.7.10.4.4

After simplifying, there are only two unique equations to be solved.

$y - 1 = \frac{x^{2} (y + 1)}{e^{- 2 C}}$

$y - 1 = - \frac{x^{2} (y + 1)}{e^{- 2 C}}$

Step 3.7.10.4.5

Solve $y - 1 = \frac{x^{2} (y + 1)}{e^{- 2 C}}$ for $y$ .

Tap for more steps...

Step 3.7.10.4.5.1

Multiply both sides by $e^{- 2 C}$ .

$(y - 1) e^{- 2 C} = \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.5.2

Simplify.

Tap for more steps...

Step 3.7.10.4.5.2.1

Simplify the left side.

Tap for more steps...

Step 3.7.10.4.5.2.1.1

Simplify $(y - 1) e^{- 2 C}$ .

Tap for more steps...

Step 3.7.10.4.5.2.1.1.1

Apply the distributive property.

$y e^{- 2 C} - 1 e^{- 2 C} = \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.5.2.1.1.2

Rewrite $- 1 e^{- 2 C}$ as $- e^{- 2 C}$ .

$y e^{- 2 C} - e^{- 2 C} = \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.5.2.2

Simplify the right side.

Tap for more steps...

Step 3.7.10.4.5.2.2.1

Simplify $\frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$ .

Tap for more steps...

Step 3.7.10.4.5.2.2.1.1

Cancel the common factor of $e^{- 2 C}$ .

Tap for more steps...

Step 3.7.10.4.5.2.2.1.1.1

Cancel the common factor.

$y e^{- 2 C} - e^{- 2 C} = \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.5.2.2.1.1.2

Rewrite the expression.

$y e^{- 2 C} - e^{- 2 C} = x^{2} (y + 1)$

Step 3.7.10.4.5.2.2.1.2

Apply the distributive property.

$y e^{- 2 C} - e^{- 2 C} = x^{2} y + x^{2} \cdot 1$

Step 3.7.10.4.5.2.2.1.3

Multiply $x^{2}$ by $1$ .

$y e^{- 2 C} - e^{- 2 C} = x^{2} y + x^{2}$

Step 3.7.10.4.5.3

Solve for $y$ .

Tap for more steps...

Step 3.7.10.4.5.3.1

Subtract $x^{2} y$ from both sides of the equation.

$y e^{- 2 C} - e^{- 2 C} - x^{2} y = x^{2}$

Step 3.7.10.4.5.3.2

Add $e^{- 2 C}$ to both sides of the equation.

$y e^{- 2 C} - x^{2} y = x^{2} + e^{- 2 C}$

Step 3.7.10.4.5.3.3

Factor $y$ out of $y e^{- 2 C} - x^{2} y$ .

Tap for more steps...

Step 3.7.10.4.5.3.3.1

Factor $y$ out of $y e^{- 2 C}$ .

$y e^{- 2 C} - x^{2} y = x^{2} + e^{- 2 C}$

Step 3.7.10.4.5.3.3.2

Factor $y$ out of $- x^{2} y$ .

$y e^{- 2 C} + y (- x^{2}) = x^{2} + e^{- 2 C}$

Step 3.7.10.4.5.3.3.3

Factor $y$ out of $y e^{- 2 C} + y (- x^{2})$ .

$y (e^{- 2 C} - x^{2}) = x^{2} + e^{- 2 C}$

Step 3.7.10.4.5.3.4

Rewrite $e^{- 2 C}$ as ${(e^{- C})}^{2}$ .

$y ({(e^{- C})}^{2} - x^{2}) = x^{2} + e^{- 2 C}$

Step 3.7.10.4.5.3.5

Factor.

Tap for more steps...

Step 3.7.10.4.5.3.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{- C}$ and $b = x$ .

$y ((e^{- C} + x) (e^{- C} - x)) = x^{2} + e^{- 2 C}$

Step 3.7.10.4.5.3.5.2

Remove unnecessary parentheses.

$y (e^{- C} + x) (e^{- C} - x) = x^{2} + e^{- 2 C}$

Step 3.7.10.4.5.3.6

Divide each term in $y (e^{- C} + x) (e^{- C} - x) = x^{2} + e^{- 2 C}$ by $(e^{- C} + x) (e^{- C} - x)$ and simplify.

Tap for more steps...

Step 3.7.10.4.5.3.6.1

Divide each term in $y (e^{- C} + x) (e^{- C} - x) = x^{2} + e^{- 2 C}$ by $(e^{- C} + x) (e^{- C} - x)$ .

$\frac{y (e^{- C} + x) (e^{- C} - x)}{(e^{- C} + x) (e^{- C} - x)} = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

Step 3.7.10.4.5.3.6.2

Simplify the left side.

Tap for more steps...

Step 3.7.10.4.5.3.6.2.1

Cancel the common factor of $e^{- C} + x$ .

Tap for more steps...

Step 3.7.10.4.5.3.6.2.1.1

Cancel the common factor.

$\frac{y (e^{- C} + x) (e^{- C} - x)}{(e^{- C} + x) (e^{- C} - x)} = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

Step 3.7.10.4.5.3.6.2.1.2

Rewrite the expression.

$\frac{y (e^{- C} - x)}{e^{- C} - x} = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

Step 3.7.10.4.5.3.6.2.2

Cancel the common factor of $e^{- C} - x$ .

Tap for more steps...

Step 3.7.10.4.5.3.6.2.2.1

Cancel the common factor.

$\frac{y (e^{- C} - x)}{e^{- C} - x} = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

Step 3.7.10.4.5.3.6.2.2.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

Step 3.7.10.4.6

Solve $y - 1 = - \frac{x^{2} (y + 1)}{e^{- 2 C}}$ for $y$ .

Tap for more steps...

Step 3.7.10.4.6.1

Multiply both sides by $e^{- 2 C}$ .

$(y - 1) e^{- 2 C} = - \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.6.2

Simplify.

Tap for more steps...

Step 3.7.10.4.6.2.1

Simplify the left side.

Tap for more steps...

Step 3.7.10.4.6.2.1.1

Simplify $(y - 1) e^{- 2 C}$ .

Tap for more steps...

Step 3.7.10.4.6.2.1.1.1

Apply the distributive property.

$y e^{- 2 C} - 1 e^{- 2 C} = - \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.6.2.1.1.2

Rewrite $- 1 e^{- 2 C}$ as $- e^{- 2 C}$ .

$y e^{- 2 C} - e^{- 2 C} = - \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.6.2.2

Simplify the right side.

Tap for more steps...

Step 3.7.10.4.6.2.2.1

Simplify $- \frac{x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$ .

Tap for more steps...

Step 3.7.10.4.6.2.2.1.1

Cancel the common factor of $e^{- 2 C}$ .

Tap for more steps...

Step 3.7.10.4.6.2.2.1.1.1

Move the leading negative in $- \frac{x^{2} (y + 1)}{e^{- 2 C}}$ into the numerator.

$y e^{- 2 C} - e^{- 2 C} = \frac{- x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.6.2.2.1.1.2

Cancel the common factor.

$y e^{- 2 C} - e^{- 2 C} = \frac{- x^{2} (y + 1)}{e^{- 2 C}} e^{- 2 C}$

Step 3.7.10.4.6.2.2.1.1.3

Rewrite the expression.

$y e^{- 2 C} - e^{- 2 C} = - x^{2} (y + 1)$

Step 3.7.10.4.6.2.2.1.2

Apply the distributive property.

$y e^{- 2 C} - e^{- 2 C} = - x^{2} y - x^{2} \cdot 1$

Step 3.7.10.4.6.2.2.1.3

Multiply $- 1$ by $1$ .

$y e^{- 2 C} - e^{- 2 C} = - x^{2} y - x^{2}$

Step 3.7.10.4.6.3

Solve for $y$ .

Tap for more steps...

Step 3.7.10.4.6.3.1

Add $x^{2} y$ to both sides of the equation.

$y e^{- 2 C} - e^{- 2 C} + x^{2} y = - x^{2}$

Step 3.7.10.4.6.3.2

Add $e^{- 2 C}$ to both sides of the equation.

$y e^{- 2 C} + x^{2} y = - x^{2} + e^{- 2 C}$

Step 3.7.10.4.6.3.3

Factor $y$ out of $y e^{- 2 C} + x^{2} y$ .

Tap for more steps...

Step 3.7.10.4.6.3.3.1

Factor $y$ out of $y e^{- 2 C}$ .

$y e^{- 2 C} + x^{2} y = - x^{2} + e^{- 2 C}$

Step 3.7.10.4.6.3.3.2

Factor $y$ out of $x^{2} y$ .

$y e^{- 2 C} + y x^{2} = - x^{2} + e^{- 2 C}$

Step 3.7.10.4.6.3.3.3

Factor $y$ out of $y e^{- 2 C} + y x^{2}$ .

$y (e^{- 2 C} + x^{2}) = - x^{2} + e^{- 2 C}$

Step 3.7.10.4.6.3.4

Divide each term in $y (e^{- 2 C} + x^{2}) = - x^{2} + e^{- 2 C}$ by $e^{- 2 C} + x^{2}$ and simplify.

Tap for more steps...

Step 3.7.10.4.6.3.4.1

Divide each term in $y (e^{- 2 C} + x^{2}) = - x^{2} + e^{- 2 C}$ by $e^{- 2 C} + x^{2}$ .

$\frac{y (e^{- 2 C} + x^{2})}{e^{- 2 C} + x^{2}} = \frac{- x^{2}}{e^{- 2 C} + x^{2}} + \frac{e^{- 2 C}}{e^{- 2 C} + x^{2}}$

Step 3.7.10.4.6.3.4.2

Simplify the left side.

Tap for more steps...

Step 3.7.10.4.6.3.4.2.1

Cancel the common factor of $e^{- 2 C} + x^{2}$ .

Tap for more steps...

Step 3.7.10.4.6.3.4.2.1.1

Cancel the common factor.

$\frac{y (e^{- 2 C} + x^{2})}{e^{- 2 C} + x^{2}} = \frac{- x^{2}}{e^{- 2 C} + x^{2}} + \frac{e^{- 2 C}}{e^{- 2 C} + x^{2}}$

Step 3.7.10.4.6.3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{e^{- 2 C} + x^{2}} + \frac{e^{- 2 C}}{e^{- 2 C} + x^{2}}$

Step 3.7.10.4.6.3.4.3

Simplify the right side.

Tap for more steps...

Step 3.7.10.4.6.3.4.3.1

Combine the numerators over the common denominator.

$y = \frac{- x^{2} + e^{- 2 C}}{e^{- 2 C} + x^{2}}$

Step 3.7.10.4.6.3.4.3.2

Simplify the numerator.

Tap for more steps...

Step 3.7.10.4.6.3.4.3.2.1

Rewrite $e^{- 2 C}$ as ${(e^{- C})}^{2}$ .

$y = \frac{- x^{2} + {(e^{- C})}^{2}}{e^{- 2 C} + x^{2}}$

Step 3.7.10.4.6.3.4.3.2.2

Reorder $- x^{2}$ and ${(e^{- C})}^{2}$ .

$y = \frac{{(e^{- C})}^{2} - x^{2}}{e^{- 2 C} + x^{2}}$

Step 3.7.10.4.6.3.4.3.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{- C}$ and $b = x$ .

$y = \frac{(e^{- C} + x) (e^{- C} - x)}{e^{- 2 C} + x^{2}}$

Step 3.7.10.4.7

List all of the solutions.

$y = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

$y = \frac{(e^{- C} + x) (e^{- C} - x)}{e^{- 2 C} + x^{2}}$

$y = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

$y = \frac{(e^{- C} + x) (e^{- C} - x)}{e^{- 2 C} + x^{2}}$

$y = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

$y = \frac{(e^{- C} + x) (e^{- C} - x)}{e^{- 2 C} + x^{2}}$

$y = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

$y = \frac{(e^{- C} + x) (e^{- C} - x)}{e^{- 2 C} + x^{2}}$

$y = \frac{x^{2}}{(e^{- C} + x) (e^{- C} - x)} + \frac{e^{- 2 C}}{(e^{- C} + x) (e^{- C} - x)}$

$y = \frac{(e^{- C} + x) (e^{- C} - x)}{e^{- 2 C} + x^{2}}$

Step 4

Simplify the constant of integration.

$y = \frac{x^{2}}{(C + x) (C - x)} + \frac{C}{(C + x) (C - x)}$

$y = \frac{(C + x) (C - x)}{C + x^{2}}$