Calculus Examples

$\frac{d y}{d x} = \frac{x^{2} y - y}{y + 1}$ , $y (3) = 1$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Factor.

Tap for more steps...

Step 1.1.1

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

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

Factor $y$ out of $- y$ .

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

Step 1.1.1.3

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

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

Step 1.1.2

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

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

Step 1.1.3

Factor.

Tap for more steps...

Step 1.1.3.1

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 $b = 1$ .

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

Step 1.1.3.2

Remove unnecessary parentheses.

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

Step 1.2

Regroup factors.

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

Step 1.3

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

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

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

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

Tap for more steps...

Step 1.4.1.1

Apply the distributive property.

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

Step 1.4.1.2

Apply the distributive property.

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

Step 1.4.1.3

Apply the distributive property.

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

Step 1.4.2

Simplify and combine like terms.

Tap for more steps...

Step 1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.1.1

Multiply $x$ by $x$ .

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

Step 1.4.2.1.2

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

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

Step 1.4.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.4.2.1.4

Multiply $x$ by $1$ .

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

Step 1.4.2.1.5

Multiply $- 1$ by $1$ .

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

Step 1.4.2.2

Add $- x$ and $x$ .

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

Step 1.4.2.3

Add $x^{2}$ and $0$ .

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

Step 1.4.3

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

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

Step 1.4.4

Cancel the common factor of $y + 1$ .

Tap for more steps...

Step 1.4.4.1

Cancel the common factor.

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

Step 1.4.4.2

Rewrite the expression.

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

Step 1.4.5

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.4.5.1

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

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

Step 1.4.5.2

Cancel the common factor.

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

Step 1.4.5.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

Split the single integral into multiple integrals.

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

Step 2.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Rewrite the expression.

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

Step 2.2.4

Apply the constant rule.

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 2.3.3

Apply the constant rule.

$y + \ln (| y |) + C_{3} = \frac{1}{3} x^{3} + C_{4} - x + C_{5}$

Step 2.3.4

Simplify.

$y + \ln (| y |) + C_{3} = \frac{1}{3} x^{3} - x + C_{6}$

Step 2.4

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

$y + \ln (| y |) = \frac{1}{3} x^{3} - x + C$

Step 3

Use the initial condition to find the value of $C$ by substituting $3$ for $x$ and $1$ for $y$ in $y + \ln (| y |) = \frac{1}{3} x^{3} - x + C$ .

$1 + \ln (| 1 |) = \frac{1}{3} \cdot 3^{3} - 1 \cdot 3 + C$

Step 4

Solve for $C$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $\frac{1}{3} \cdot 3^{3} - 1 \cdot 3 + C = 1 + \ln (| 1 |)$ .

$\frac{1}{3} \cdot 3^{3} - 1 \cdot 3 + C = 1 + \ln (| 1 |)$

Step 4.2

Simplify $\frac{1}{3} \cdot 3^{3} - 1 \cdot 3 + C$ .

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1.1.1

Factor $3$ out of $3^{3}$ .

$\frac{1}{3} \cdot (3 \cdot 3^{2}) - 1 \cdot 3 + C = 1 + \ln (| 1 |)$

Step 4.2.1.1.2

Cancel the common factor.

$\frac{1}{3} \cdot (3 \cdot 3^{2}) - 1 \cdot 3 + C = 1 + \ln (| 1 |)$

Step 4.2.1.1.3

Rewrite the expression.

$3^{2} - 1 \cdot 3 + C = 1 + \ln (| 1 |)$

Step 4.2.1.2

Raise $3$ to the power of $2$ .

$9 - 1 \cdot 3 + C = 1 + \ln (| 1 |)$

Step 4.2.1.3

Multiply $- 1$ by $3$ .

$9 - 3 + C = 1 + \ln (| 1 |)$

Step 4.2.2

Subtract $3$ from $9$ .

$6 + C = 1 + \ln (| 1 |)$

Step 4.3

Simplify $1 + \ln (| 1 |)$ .

Tap for more steps...

Step 4.3.1

Simplify each term.

Tap for more steps...

Step 4.3.1.1

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$6 + C = 1 + \ln (1)$

Step 4.3.1.2

The natural logarithm of $1$ is $0$ .

$6 + C = 1 + 0$

Step 4.3.2

Add $1$ and $0$ .

$6 + C = 1$

Step 4.4

Move all terms not containing $C$ to the right side of the equation.

Tap for more steps...

Step 4.4.1

Subtract $6$ from both sides of the equation.

$C = 1 - 6$

Step 4.4.2

Subtract $6$ from $1$ .

$C = - 5$

Step 5

Substitute $- 5$ for $C$ in $y + \ln (| y |) = \frac{1}{3} x^{3} - x + C$ and simplify.

Tap for more steps...

Step 5.1

Substitute $- 5$ for $C$ .

$y + \ln (| y |) = \frac{1}{3} x^{3} - x - 5$

Step 5.2

Combine $\frac{1}{3}$ and $x^{3}$ .

$y + \ln (| y |) = \frac{x^{3}}{3} - x - 5$