Calculus Examples

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

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $x^{2} - 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Factor $y$ out of $\frac{2 y}{x^{2} - 1}$ .

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

Step 1.4

Reorder $y$ and $\frac{2}{x^{2} - 1}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{2}{x^{2} - 1}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{2}{x^{2} - 1} d x}$

Step 2.2

Integrate $\frac{2}{x^{2} - 1}$ .

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Step 2.2.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{2 \int \frac{1}{x^{2} - 1} d x}$

Step 2.2.2

Write the fraction using partial fraction decomposition.

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Step 2.2.2.1

Decompose the fraction and multiply through by the common denominator.

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Step 2.2.2.1.1

Factor the fraction.

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Step 2.2.2.1.1.1

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

$\frac{1}{x^{2} - 1^{2}}$

Step 2.2.2.1.1.2

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

$\frac{1}{(x + 1) (x - 1)}$

Step 2.2.2.1.2

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 $A$ .

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

Step 2.2.2.1.3

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

Step 2.2.2.1.4

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

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

Step 2.2.2.1.5

Cancel the common factor of $x + 1$ .

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Step 2.2.2.1.5.1

Cancel the common factor.

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

Step 2.2.2.1.5.2

Rewrite the expression.

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

Step 2.2.2.1.6

Cancel the common factor of $x - 1$ .

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Step 2.2.2.1.6.1

Cancel the common factor.

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

Step 2.2.2.1.6.2

Rewrite the expression.

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

Step 2.2.2.1.7

Simplify each term.

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Step 2.2.2.1.7.1

Cancel the common factor of $x + 1$ .

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Step 2.2.2.1.7.1.1

Cancel the common factor.

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

Step 2.2.2.1.7.1.2

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

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

Step 2.2.2.1.7.2

Apply the distributive property.

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

Step 2.2.2.1.7.3

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

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

Step 2.2.2.1.7.4

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

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

Step 2.2.2.1.7.5

Cancel the common factor of $x - 1$ .

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Step 2.2.2.1.7.5.1

Cancel the common factor.

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

Step 2.2.2.1.7.5.2

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

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

Step 2.2.2.1.7.6

Apply the distributive property.

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

Step 2.2.2.1.7.7

Multiply $B$ by $1$ .

$1 = A x - A + B x + B$

Step 2.2.2.1.8

Move $- A$ .

$1 = A x + B x - A + B$

Step 2.2.2.2

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

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Step 2.2.2.2.1

Create an equation for the partial fraction variables by equating the coefficients of $x$ 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 2.2.2.2.2

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

$1 = - 1 A + B$

Step 2.2.2.2.3

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 2.2.2.3

Solve the system of equations.

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Step 2.2.2.3.1

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

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Step 2.2.2.3.1.1

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

$A + B = 0$

$1 = - 1 A + B$

Step 2.2.2.3.1.2

Subtract $B$ from both sides of the equation.

$A = - B$

$1 = - 1 A + B$

$A = - B$

$1 = - 1 A + B$

Step 2.2.2.3.2

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

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Step 2.2.2.3.2.1

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

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

$A = - B$

Step 2.2.2.3.2.2

Simplify the right side.

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Step 2.2.2.3.2.2.1

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

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Step 2.2.2.3.2.2.1.1

Multiply $- 1 (- B)$ .

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Step 2.2.2.3.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$1 = 1 B + B$

$A = - B$

Step 2.2.2.3.2.2.1.1.2

Multiply $B$ by $1$ .

$1 = B + B$

$A = - B$

$1 = B + B$

$A = - B$

Step 2.2.2.3.2.2.1.2

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 2.2.2.3.3

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

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Step 2.2.2.3.3.1

Rewrite the equation as $2 B = 1$ .

$2 B = 1$

$A = - B$

Step 2.2.2.3.3.2

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

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Step 2.2.2.3.3.2.1

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

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

$A = - B$

Step 2.2.2.3.3.2.2

Simplify the left side.

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Step 2.2.2.3.3.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.3.3.2.2.1.1

Cancel the common factor.

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

$A = - B$

Step 2.2.2.3.3.2.2.1.2

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 2.2.2.3.4

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

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Step 2.2.2.3.4.1

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

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

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

Step 2.2.2.3.4.2

Simplify the right side.

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Step 2.2.2.3.4.2.1

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 2.2.2.3.5

List all of the solutions.

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

Step 2.2.2.4

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

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

Step 2.2.2.5

Simplify.

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Step 2.2.2.5.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2.2.5.2

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

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

Step 2.2.2.5.3

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

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

Step 2.2.2.5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2.2.5.5

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

$e^{2 \int - \frac{1}{2 (x + 1)} + \frac{1}{2 (x - 1)} d x}$

Step 2.2.3

Split the single integral into multiple integrals.

$e^{2 (\int - \frac{1}{2 (x + 1)} d x + \int \frac{1}{2 (x - 1)} d x)}$

Step 2.2.4

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

$e^{2 (- \int \frac{1}{2 (x + 1)} d x + \int \frac{1}{2 (x - 1)} d x)}$

Step 2.2.5

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

$e^{2 (- (\frac{1}{2} \int \frac{1}{x + 1} d x) + \int \frac{1}{2 (x - 1)} d x)}$

Step 2.2.6

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

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Step 2.2.6.1

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

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Step 2.2.6.1.1

Differentiate $x + 1$ .

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

Step 2.2.6.1.2

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

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

Step 2.2.6.1.3

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

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

Step 2.2.6.1.4

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

$1 + 0$

Step 2.2.6.1.5

Add $1$ and $0$ .

$1$

Step 2.2.6.2

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

$e^{2 (- \frac{1}{2} \int \frac{1}{u_{1}} d u_{1} + \int \frac{1}{2 (x - 1)} d x)}$

Step 2.2.7

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

$e^{2 (- \frac{1}{2} (\ln (| u_{1} |) + C) + \int \frac{1}{2 (x - 1)} d x)}$

Step 2.2.8

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

$e^{2 (- \frac{1}{2} (\ln (| u_{1} |) + C) + \frac{1}{2} \int \frac{1}{x - 1} d x)}$

Step 2.2.9

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

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Step 2.2.9.1

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

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Step 2.2.9.1.1

Differentiate $x - 1$ .

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

Step 2.2.9.1.2

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

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

Step 2.2.9.1.3

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

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

Step 2.2.9.1.4

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

$1 + 0$

Step 2.2.9.1.5

Add $1$ and $0$ .

$1$

Step 2.2.9.2

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

$e^{2 (- \frac{1}{2} (\ln (| u_{1} |) + C) + \frac{1}{2} \int \frac{1}{u_{2}} d u_{2})}$

Step 2.2.10

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

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

Step 2.2.11

Simplify.

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

Step 2.2.12

Substitute back in for each integration substitution variable.

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Step 2.2.12.1

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

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

Step 2.2.12.2

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

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

Step 2.2.13

Simplify.

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Step 2.2.13.1

Simplify each term.

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Step 2.2.13.1.1

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

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

Step 2.2.13.1.2

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

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

Step 2.2.13.2

Combine the numerators over the common denominator.

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

Step 2.2.13.3

Cancel the common factor of $2$ .

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Step 2.2.13.3.1

Cancel the common factor.

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

Step 2.2.13.3.2

Rewrite the expression.

$e^{- \ln (| x + 1 |) + \ln (| x - 1 |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- \ln (x + 1) + \ln (x - 1)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln ({(x + 1)}^{- 1}) + \ln (x - 1)}$

Step 2.5

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$e^{\ln ({(x + 1)}^{- 1} (x - 1))}$

Step 2.6

Exponentiation and log are inverse functions.

${(x + 1)}^{- 1} (x - 1)$

Step 2.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x + 1} (x - 1)$

Step 2.8

Multiply $\frac{1}{x + 1}$ by $x - 1$ .

$\frac{x - 1}{x + 1}$

Step 3

Multiply each term by the integrating factor $\frac{x - 1}{x + 1}$ .

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Step 3.1

Multiply each term by $\frac{x - 1}{x + 1}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{x - 1}{x + 1}$ and $\frac{d y}{d x}$ .

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

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.3

Combine $\frac{2}{(x + 1) (x - 1)}$ and $y$ .

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

Step 3.2.4

Cancel the common factor of $x - 1$ .

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Step 3.2.4.1

Factor $x - 1$ out of $(x + 1) (x - 1)$ .

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

Step 3.2.4.2

Cancel the common factor.

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

Step 3.2.4.3

Rewrite the expression.

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

Step 3.2.5

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

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

Step 3.2.6

Raise $x + 1$ to the power of $1$ .

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

Step 3.2.7

Raise $x + 1$ to the power of $1$ .

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

Step 3.2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.9

Add $1$ and $1$ .

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

Step 3.3

To write $\frac{(x - 1) \frac{d y}{d x}}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 3.4

Write each expression with a common denominator of ${(x + 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{(x - 1) \frac{d y}{d x}}{x + 1}$ by $\frac{x + 1}{x + 1}$ .

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

Step 3.4.2

Raise $x + 1$ to the power of $1$ .

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

Step 3.4.3

Raise $x + 1$ to the power of $1$ .

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

Step 3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Rewrite $- 1 \frac{d y}{d x}$ as $- \frac{d y}{d x}$ .

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

Step 3.6.3

Expand $(x \frac{d y}{d x} - \frac{d y}{d x}) (x + 1)$ using the FOIL Method.

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Step 3.6.3.1

Apply the distributive property.

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

Step 3.6.3.2

Apply the distributive property.

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

Step 3.6.3.3

Apply the distributive property.

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

Step 3.6.4

Simplify and combine like terms.

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Step 3.6.4.1

Simplify each term.

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Step 3.6.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.6.4.1.1.1

Move $x$ .

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

Step 3.6.4.1.1.2

Multiply $x$ by $x$ .

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

Step 3.6.4.1.2

Multiply $x$ by $1$ .

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

Step 3.6.4.1.3

Multiply $- 1$ by $1$ .

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

Step 3.6.4.2

Subtract $\frac{d y}{d x} x$ from $x \frac{d y}{d x}$ .

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Step 3.6.4.2.1

Move $\frac{d y}{d x}$ .

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

Step 3.6.4.2.2

Subtract $x \frac{d y}{d x}$ from $x \frac{d y}{d x}$ .

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

Step 3.6.4.3

Add $x^{2} \frac{d y}{d x}$ and $0$ .

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

Step 3.7

Cancel the common factor of $x + 1$ .

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Step 3.7.1

Factor $x + 1$ out of ${(x + 1)}^{2}$ .

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

Step 3.7.2

Cancel the common factor.

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

Step 3.7.3

Rewrite the expression.

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

Step 3.8

Simplify the denominator.

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Step 3.8.1

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

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

Step 3.8.2

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

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

Step 3.9

Cancel the common factor of $x - 1$ .

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Step 3.9.1

Factor $x - 1$ out of $(x + 1) (x - 1)$ .

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

Step 3.9.2

Cancel the common factor.

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

Step 3.9.3

Rewrite the expression.

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

Step 3.10

Cancel the common factor of $x + 1$ .

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Step 3.10.1

Cancel the common factor.

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

Step 3.10.2

Rewrite the expression.

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

Step 4

Rewrite the left side as a result of differentiating a product.

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Apply the constant rule.

$\frac{x - 1}{x + 1} y = x + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $\frac{x - 1}{x + 1} y$ .

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.2

Multiply both sides by $x + 1$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Simplify $\frac{y (x - 1)}{x + 1} (x + 1)$ .

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Step 8.3.1.1.1

Simplify terms.

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Step 8.3.1.1.1.1

Cancel the common factor of $x + 1$ .

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Step 8.3.1.1.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.1.1.2

Rewrite the expression.

$y (x - 1) = (x + C) (x + 1)$

Step 8.3.1.1.1.2

Apply the distributive property.

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

Step 8.3.1.1.1.3

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

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

Step 8.3.1.1.2

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

$y x - y = (x + C) (x + 1)$

Step 8.3.1.1.3

Reorder $y$ and $x$ .

$x y - y = (x + C) (x + 1)$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(x + C) (x + 1)$ .

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Step 8.3.2.1.1

Expand $(x + C) (x + 1)$ using the FOIL Method.

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Step 8.3.2.1.1.1

Apply the distributive property.

$x y - y = x (x + 1) + C (x + 1)$

Step 8.3.2.1.1.2

Apply the distributive property.

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

Step 8.3.2.1.1.3

Apply the distributive property.

$x y - y = x \cdot x + x \cdot 1 + C x + C \cdot 1$

Step 8.3.2.1.2

Simplify terms.

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Step 8.3.2.1.2.1

Simplify each term.

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Step 8.3.2.1.2.1.1

Multiply $x$ by $x$ .

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

Step 8.3.2.1.2.1.2

Multiply $x$ by $1$ .

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

Step 8.3.2.1.2.1.3

Multiply $C$ by $1$ .

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

Step 8.3.2.1.2.2

Simplify the expression.

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Step 8.3.2.1.2.2.1

Move $x$ .

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

Step 8.3.2.1.2.2.2

Reorder $x^{2}$ and $C x$ .

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

Step 8.4

Solve for $y$ .

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Step 8.4.1

Factor $y$ out of $x y - y$ .

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Step 8.4.1.1

Factor $y$ out of $x y$ .

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

Step 8.4.1.2

Factor $y$ out of $- y$ .

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

Step 8.4.1.3

Factor $y$ out of $y x + y \cdot - 1$ .

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

Step 8.4.2

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

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Step 8.4.2.1

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

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

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $x - 1$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

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

Step 8.4.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Simplify terms.

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Step 8.4.2.3.1.1

Combine the numerators over the common denominator.

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

Step 8.4.2.3.1.2

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

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Step 8.4.2.3.1.2.1

Factor $x$ out of $C x$ .

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

Step 8.4.2.3.1.2.2

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

$y = \frac{x C + x \cdot x}{x - 1} + \frac{C}{x - 1} + \frac{x}{x - 1}$

Step 8.4.2.3.1.2.3

Factor $x$ out of $x C + x \cdot x$ .

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

Step 8.4.2.3.1.3

Combine the numerators over the common denominator.

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

Step 8.4.2.3.2

Simplify the numerator.

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Step 8.4.2.3.2.1

Apply the distributive property.

$y = \frac{x C + x \cdot x + C}{x - 1} + \frac{x}{x - 1}$

Step 8.4.2.3.2.2

Multiply $x$ by $x$ .

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

Step 8.4.2.3.3

Combine the numerators over the common denominator.

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

Step 8.4.2.3.4

Simplify the numerator.

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Step 8.4.2.3.4.1

Factor out the greatest common factor from each group.

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Step 8.4.2.3.4.1.1

Group the first two terms and the last two terms.

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

Step 8.4.2.3.4.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 8.4.2.3.4.2

Factor the polynomial by factoring out the greatest common factor, $C + x$ .

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