Calculus Examples

$2 x y y' = y^{2} - 2 x^{3}$

Step 1

Rewrite the differential equation.

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

Step 2

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 3

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

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Step 3.1

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 $g (x) = y$ .

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Step 3.1.1

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 3.1.2

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 3.1.3

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

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

Step 3.2

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

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

Step 4

Substitute the derivative back in to the differential equation.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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Step 5.1

Subtract $v$ from both sides of the equation.

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

Step 5.2

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 5.3

Cancel the common factor of $x$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

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

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

Step 5.4.2.2

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

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

Step 5.4.2.3

Cancel the common factor.

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

Step 5.4.2.4

Rewrite the expression.

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

Step 5.4.2.5

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 6

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

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Step 6.1

Set up the integration.

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

Step 6.2

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

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Step 6.2.1

Split the fraction into multiple fractions.

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

Simplify.

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

Step 6.3

Remove the constant of integration.

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

Step 6.4

Use the logarithmic power rule.

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

Step 6.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 6.6

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

$\frac{1}{x}$

Step 7

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

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Step 7.1

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 7.2

Simplify each term.

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Step 7.2.1

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 7.2.2

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 7.2.3

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 7.2.4

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 7.2.5

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

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Step 7.2.5.1

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 7.2.5.2

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 7.2.5.3

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 7.2.5.4

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 7.2.5.5

Add $1$ and $1$ .

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

Step 7.3

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 7.4

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

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

Step 7.5

Cancel the common factor of $x$ .

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Step 7.5.1

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

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

Step 7.5.2

Cancel the common factor.

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

Step 7.5.3

Rewrite the expression.

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

Step 8

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

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

Step 9

Set up an integral on each side.

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

Step 10

Integrate the left side.

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

Step 11

Integrate the right side.

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Step 11.1

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

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

Step 11.2

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 11.3

Simplify the answer.

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Step 11.3.1

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 11.3.2

Simplify.

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Step 11.3.2.1

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

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

Step 11.3.2.2

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

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Step 11.3.2.2.1

Factor $2$ out of $- 2$ .

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

Step 11.3.2.2.2

Cancel the common factors.

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Step 11.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 11.3.2.2.2.2

Cancel the common factor.

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

Step 11.3.2.2.2.3

Rewrite the expression.

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

Step 11.3.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 12

Solve for $v$ .

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Step 12.1

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

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

Step 12.2

Multiply both sides by $x$ .

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

Step 12.3

Simplify.

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Step 12.3.1

Simplify the left side.

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Step 12.3.1.1

Cancel the common factor of $x$ .

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Step 12.3.1.1.1

Cancel the common factor.

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

Step 12.3.1.1.2

Rewrite the expression.

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

Step 12.3.2

Simplify the right side.

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Step 12.3.2.1

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

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Step 12.3.2.1.1

Apply the distributive property.

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

Step 12.3.2.1.2

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

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Step 12.3.2.1.2.1

Move $x$ .

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

Step 12.3.2.1.2.2

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

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Step 12.3.2.1.2.2.1

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

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

Step 12.3.2.1.2.2.2

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

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

Step 12.3.2.1.2.3

Add $1$ and $2$ .

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

Step 13

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

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

Step 14

Solve for $y$ .

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Step 14.1

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 14.2

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

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Step 14.2.1

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

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

Step 14.2.2

Factor $x$ out of $C x$ .

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

Step 14.2.3

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

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

Step 14.3

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

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Step 14.3.1

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

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

Step 14.3.2

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

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

Step 14.3.3

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

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

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

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

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

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

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