Calculus Examples

$2 x y \frac{d y}{d x} = y^{2} - 2 x^{3}$

Step 1

Let $v = y^{2}$ . Substitute $v$ for all occurrences of $y^{2}$ .

$2 x y \frac{d y}{d x} = v - 2 x^{3}$

Step 2

Find $\frac{d v}{d x}$ by differentiating $y^{2}$ .

Tap for more steps...

Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d v}{d x} = \frac{d}{d u} [u^{2}] \frac{d}{d x} [y]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$\frac{d v}{d x} = 2 u \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

$\frac{d v}{d x} = 2 y \frac{d}{d x} [y]$

Step 2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{d v}{d x} = 2 y y'$

Step 3

Substitute the derivative back in to the differential equation.

Tap for more steps...

Step 3.1

Factor $2 y$ out of $2 x y$ .

$2 y (x) \frac{\frac{d v}{d x}}{2 y} = v - 2 x^{3}$

Step 3.2

Cancel the common factor.

$2 y x \frac{\frac{d v}{d x}}{2 y} = v - 2 x^{3}$

Step 3.3

Rewrite the expression.

$x \frac{d v}{d x} = v - 2 x^{3}$

Step 4

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

Tap for more steps...

Step 4.1

Subtract $v$ from both sides of the equation.

$x \frac{d v}{d x} - v = - 2 x^{3}$

Step 4.2

Divide each term in $x \frac{d v}{d x} - v = - 2 x^{3}$ by $x$ .

$\frac{x \frac{d v}{d x}}{x} + \frac{- v}{x} = \frac{- 2 x^{3}}{x}$

Step 4.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.3.1

Cancel the common factor.

$\frac{x \frac{d v}{d x}}{x} + \frac{- v}{x} = \frac{- 2 x^{3}}{x}$

Step 4.3.2

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

$\frac{d v}{d x} + \frac{- v}{x} = \frac{- 2 x^{3}}{x}$

Step 4.4

Cancel the common factor of $x^{3}$ and $x$ .

Tap for more steps...

Step 4.4.1

Factor $x$ out of $- 2 x^{3}$ .

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

Step 4.4.2

Cancel the common factors.

Tap for more steps...

Step 4.4.2.1

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

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

Step 4.4.2.2

Factor $x$ out of $x^{1}$ .

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

Step 4.4.2.3

Cancel the common factor.

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

Step 4.4.2.4

Rewrite the expression.

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

Step 4.4.2.5

Divide $- 2 x^{2}$ by $1$ .

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

Step 4.5

Factor $v$ out of $\frac{- v}{x}$ .

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

Step 4.6

Reorder $v$ and $\frac{- 1}{x}$ .

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

Step 5

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

Tap for more steps...

Step 5.1

Set up the integration.

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Split the fraction into multiple fractions.

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Simplify.

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

Step 5.3

Remove the constant of integration.

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

Step 5.4

Use the logarithmic power rule.

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

Step 5.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 5.6

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

$\frac{1}{x}$

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify each term.

Tap for more steps...

Step 6.2.1

Combine $\frac{1}{x}$ and $\frac{d v}{d x}$ .

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

Step 6.2.2

Move the negative in front of the fraction.

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

Step 6.2.3

Rewrite using the commutative property of multiplication.

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

Step 6.2.4

Combine $\frac{1}{x}$ and $v$ .

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

Step 6.2.5

Multiply $- \frac{1}{x} \cdot \frac{v}{x}$ .

Tap for more steps...

Step 6.2.5.1

Multiply $\frac{v}{x}$ by $\frac{1}{x}$ .

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

Step 6.2.5.2

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

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

Step 6.2.5.3

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

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

Step 6.2.5.4

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

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

Step 6.2.5.5

Add $1$ and $1$ .

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

Step 6.3

Rewrite using the commutative property of multiplication.

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

Step 6.4

Combine $- 2$ and $\frac{1}{x}$ .

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = \frac{- 2}{x} x^{2}$

Step 6.5

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.5.1

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

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = \frac{- 2}{x} (x \cdot x)$

Step 6.5.2

Cancel the common factor.

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = \frac{- 2}{x} (x \cdot x)$

Step 6.5.3

Rewrite the expression.

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = - 2 x$

Step 7

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

$\frac{d}{d x} [\frac{1}{x} v] = - 2 x$

Step 8

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} v] d x = \int - 2 x d x$

Step 9

Integrate the left side.

$\frac{1}{x} v = \int - 2 x d x$

Step 10

Integrate the right side.

Tap for more steps...

Step 10.1

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

$\frac{1}{x} v = - 2 \int x d x$

Step 10.2

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

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

Step 10.3

Simplify the answer.

Tap for more steps...

Step 10.3.1

Rewrite $- 2 (\frac{1}{2} x^{2} + C)$ as $- 2 (\frac{1}{2}) x^{2} + C$ .

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

Step 10.3.2

Simplify.

Tap for more steps...

Step 10.3.2.1

Combine $- 2$ and $\frac{1}{2}$ .

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

Step 10.3.2.2

Cancel the common factor of $- 2$ and $2$ .

Tap for more steps...

Step 10.3.2.2.1

Factor $2$ out of $- 2$ .

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

Step 10.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 10.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 10.3.2.2.2.2

Cancel the common factor.

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

Step 10.3.2.2.2.3

Rewrite the expression.

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

Step 10.3.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 11

Solve for $v$ .

Tap for more steps...

Step 11.1

Combine $\frac{1}{x}$ and $v$ .

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

Step 11.2

Multiply both sides by $x$ .

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

Step 11.3

Simplify.

Tap for more steps...

Step 11.3.1

Simplify the left side.

Tap for more steps...

Step 11.3.1.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 11.3.1.1.1

Cancel the common factor.

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

Step 11.3.1.1.2

Rewrite the expression.

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

Step 11.3.2

Simplify the right side.

Tap for more steps...

Step 11.3.2.1

Simplify $(- x^{2} + C) x$ .

Tap for more steps...

Step 11.3.2.1.1

Apply the distributive property.

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

Step 11.3.2.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 11.3.2.1.2.1

Move $x$ .

$v = - (x \cdot x^{2}) + C x$

Step 11.3.2.1.2.2

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

Tap for more steps...

Step 11.3.2.1.2.2.1

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

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

Step 11.3.2.1.2.2.2

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

$v = - x^{1 + 2} + C x$

Step 11.3.2.1.2.3

Add $1$ and $2$ .

$v = - x^{3} + C x$

Step 12

Replace all occurrences of $v$ with $y^{2}$ .

$y^{2} = - x^{3} + C x$

Step 13

Solve for $y$ .

Tap for more steps...

Step 13.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- x^{3} + C x}$

Step 13.2

Factor $x$ out of $- x^{3} + C x$ .

Tap for more steps...

Step 13.2.1

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

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

Step 13.2.2

Factor $x$ out of $C x$ .

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

Step 13.2.3

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

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

Step 13.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 13.3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 13.3.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 13.3.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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