Calculus Examples

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

Step 1

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{3}$ .

$v = y^{- 2}$

Step 2

Solve the equation for $y$ .

$y = v^{- \frac{1}{2}}$

Step 3

Take the derivative of $y$ with respect to $x$ .

$y' = v^{- \frac{1}{2}}$

Step 4

Take the derivative of $v^{- \frac{1}{2}}$ with respect to $x$ .

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Step 4.1

Take the derivative of $v^{- \frac{1}{2}}$ .

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

Step 4.2

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

$y' = \frac{d}{d x} [\frac{1}{v^{\frac{1}{2}}}]$

Step 4.3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1$ and $g (x) = v^{\frac{1}{2}}$ .

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{{(v^{\frac{1}{2}})}^{2}}$

Step 4.4.2

Multiply the exponents in ${(v^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{v^{\frac{1}{2} \cdot 2}}$

Step 4.4.2.2

Cancel the common factor of $2$ .

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Step 4.4.2.2.1

Cancel the common factor.

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{v^{\frac{1}{2} \cdot 2}}$

Step 4.4.2.2.2

Rewrite the expression.

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

Step 4.5

Simplify.

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

Step 4.6

Differentiate using the Constant Rule.

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Step 4.6.1

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

$y' = \frac{v^{\frac{1}{2}} \cdot 0 - \frac{d}{d x} [v^{\frac{1}{2}}]}{v}$

Step 4.6.2

Simplify the expression.

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Step 4.6.2.1

Multiply $v^{\frac{1}{2}}$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v^{\frac{1}{2}}]}{v}$

Step 4.6.2.2

Subtract $\frac{d}{d x} [v^{\frac{1}{2}}]$ from $0$ .

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

Step 4.6.2.3

Move the negative in front of the fraction.

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

Step 4.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = v$ .

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Step 4.7.1

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

$y' = - \frac{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [v]}{v}$

Step 4.7.2

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

$y' = - \frac{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [v]}{v}$

Step 4.7.3

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

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

Step 4.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{\frac{1}{2} v^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [v]}{v}$

Step 4.9

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

$y' = - \frac{\frac{1}{2} v^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [v]}{v}$

Step 4.10

Combine the numerators over the common denominator.

$y' = - \frac{\frac{1}{2} v^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [v]}{v}$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Multiply $- 1$ by $2$ .

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

Step 4.11.2

Subtract $2$ from $1$ .

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

Step 4.12

Move the negative in front of the fraction.

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

Step 4.13

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

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

Step 4.14

Move $v^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.15

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

$y' = - \frac{\frac{1}{2 v^{\frac{1}{2}}} v'}{v}$

Step 4.16

Combine $\frac{1}{2 v^{\frac{1}{2}}}$ and $v'$ .

$y' = - \frac{\frac{v'}{2 v^{\frac{1}{2}}}}{v}$

Step 4.17

Rewrite $\frac{\frac{v'}{2 v^{\frac{1}{2}}}}{v}$ as a product.

$y' = - (\frac{v'}{2 v^{\frac{1}{2}}} \cdot \frac{1}{v})$

Step 4.18

Multiply $\frac{v'}{2 v^{\frac{1}{2}}}$ by $\frac{1}{v}$ .

$y' = - \frac{v'}{2 v^{\frac{1}{2}} v}$

Step 4.19

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

$y' = - \frac{v'}{2 (v^{1} v^{\frac{1}{2}})}$

Step 4.20

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

$y' = - \frac{v'}{2 v^{1 + \frac{1}{2}}}$

Step 4.21

Write $1$ as a fraction with a common denominator.

$y' = - \frac{v'}{2 v^{\frac{2}{2} + \frac{1}{2}}}$

Step 4.22

Combine the numerators over the common denominator.

$y' = - \frac{v'}{2 v^{\frac{2 + 1}{2}}}$

Step 4.23

Add $2$ and $1$ .

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

Step 5

Substitute $- \frac{v'}{2 v^{\frac{3}{2}}}$ for $\frac{d y}{d x}$ and $v^{- \frac{1}{2}}$ for $y$ in the original equation $\frac{d y}{d x} - y = x y^{3}$ .

$- \frac{v'}{2 v^{\frac{3}{2}}} - v^{- \frac{1}{2}} = x {(v^{- \frac{1}{2}})}^{3}$

Step 6

Solve the substituted differential equation.

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Step 6.1

Multiply each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - v^{- \frac{1}{2}} = x {(v^{- \frac{1}{2}})}^{3}$ by $- (2 v^{\frac{3}{2}})$ to eliminate the fractions.

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Step 6.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - v^{- \frac{1}{2}} = x {(v^{- \frac{1}{2}})}^{3}$ by $- (2 v^{\frac{3}{2}})$ .

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} (- (2 v^{\frac{3}{2}})) - v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Cancel the common factor of $2 v^{\frac{3}{2}}$ .

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Step 6.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (- (2 v^{\frac{3}{2}})) - v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.1.2

Factor $2 v^{\frac{3}{2}}$ out of $- (2 v^{\frac{3}{2}})$ .

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (2 v^{\frac{3}{2}} (- 1)) - v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (2 v^{\frac{3}{2}} \cdot - 1) - v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 - v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} - v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.3

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

$\frac{d v}{d x} - v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4

Multiply $v^{- \frac{1}{2}}$ by $v^{\frac{3}{2}}$ by adding the exponents.

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Step 6.1.2.1.4.1

Move $v^{\frac{3}{2}}$ .

$\frac{d v}{d x} - (v^{\frac{3}{2}} v^{- \frac{1}{2}}) (- 1 \cdot (2)) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.2

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

$\frac{d v}{d x} - v^{\frac{3}{2} - \frac{1}{2}} (- 1 \cdot (2)) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} - v^{\frac{3 - 1}{2}} (- 1 \cdot (2)) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.4

Subtract $1$ from $3$ .

$\frac{d v}{d x} - v^{\frac{2}{2}} (- 1 \cdot (2)) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.5

Divide $2$ by $2$ .

$\frac{d v}{d x} - v^{1} (- 1 \cdot (2)) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.5

Simplify $- v^{1} (- 1 \cdot (2))$ .

$\frac{d v}{d x} - v (- 1 \cdot (2)) = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.6

Multiply $- 1$ by $2$ .

$\frac{d v}{d x} - v \cdot - 2 = x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.7

Multiply $- 2$ by $- 1$ .

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

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.3.2

Multiply $- 1$ by $2$ .

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

Step 6.1.3.3

Multiply the exponents in ${(v^{- \frac{1}{2}})}^{3}$ .

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Step 6.1.3.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d v}{d x} + 2 v = - 2 x v^{- \frac{1}{2} \cdot 3} v^{\frac{3}{2}}$

Step 6.1.3.3.2

Multiply $- \frac{1}{2} \cdot 3$ .

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Step 6.1.3.3.2.1

Multiply $3$ by $- 1$ .

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

Step 6.1.3.3.2.2

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 6.1.3.3.3

Move the negative in front of the fraction.

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

Step 6.1.3.4

Multiply $v^{- \frac{3}{2}}$ by $v^{\frac{3}{2}}$ by adding the exponents.

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Step 6.1.3.4.1

Move $v^{\frac{3}{2}}$ .

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

Step 6.1.3.4.2

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

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

Step 6.1.3.4.3

Combine the numerators over the common denominator.

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

Step 6.1.3.4.4

Subtract $3$ from $3$ .

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

Step 6.1.3.4.5

Divide $0$ by $2$ .

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

Step 6.1.3.5

Simplify $- 2 x v^{0}$ .

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

Step 6.2

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

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Step 6.2.1

Set up the integration.

$e^{\int 2 d x}$

Step 6.2.2

Apply the constant rule.

$e^{2 x + C}$

Step 6.2.3

Remove the constant of integration.

$e^{2 x}$

Step 6.3

Multiply each term by the integrating factor $e^{2 x}$ .

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Step 6.3.1

Multiply each term by $e^{2 x}$ .

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

Step 6.3.2

Rewrite using the commutative property of multiplication.

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

Step 6.3.3

Rewrite using the commutative property of multiplication.

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

Step 6.3.4

Reorder factors in $e^{2 x} \frac{d v}{d x} + 2 e^{2 x} v = - 2 e^{2 x} x$ .

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

Step 6.4

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

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

Step 6.5

Set up an integral on each side.

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

Step 6.6

Integrate the left side.

$e^{2 x} v = \int - 2 x e^{2 x} d x$

Step 6.7

Integrate the right side.

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Step 6.7.1

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

$e^{2 x} v = - 2 \int x e^{2 x} d x$

Step 6.7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

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

Step 6.7.3

Simplify.

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Step 6.7.3.1

Combine $\frac{1}{2}$ and $e^{2 x}$ .

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

Step 6.7.3.2

Combine $x$ and $\frac{e^{2 x}}{2}$ .

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

Step 6.7.3.3

Combine $\frac{1}{2}$ and $e^{2 x}$ .

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

Step 6.7.4

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

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

Step 6.7.5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.7.5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 6.7.5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 6.7.5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 6.7.5.1.3

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

$2 \cdot 1$

Step 6.7.5.1.4

Multiply $2$ by $1$ .

$2$

Step 6.7.5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 6.7.6

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 6.7.7

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

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

Step 6.7.8

Simplify.

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Step 6.7.8.1

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

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

Step 6.7.8.2

Multiply $2$ by $2$ .

$e^{2 x} v = - 2 (\frac{x e^{2 x}}{2} - \frac{1}{4} \int e^{u} d u)$

Step 6.7.9

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{2 x} v = - 2 (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C))$

Step 6.7.10

Rewrite $- 2 (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C))$ as $- 2 (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u}) + C$ .

$e^{2 x} v = - 2 (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u}) + C$

Step 6.7.11

Replace all occurrences of $u$ with $2 x$ .

$e^{2 x} v = - 2 (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x}) + C$

Step 6.7.12

Simplify.

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Step 6.7.12.1

Simplify each term.

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Step 6.7.12.1.1

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

$e^{2 x} v = - 2 (\frac{x}{2} e^{2 x} - \frac{1}{4} e^{2 x}) + C$

Step 6.7.12.1.2

Combine $\frac{x}{2}$ and $e^{2 x}$ .

$e^{2 x} v = - 2 (\frac{x e^{2 x}}{2} - \frac{1}{4} e^{2 x}) + C$

Step 6.7.12.1.3

Combine $e^{2 x}$ and $\frac{1}{4}$ .

$e^{2 x} v = - 2 (\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4}) + C$

Step 6.7.12.2

Apply the distributive property.

$e^{2 x} v = - 2 \frac{x e^{2 x}}{2} - 2 (- \frac{e^{2 x}}{4}) + C$

Step 6.7.12.3

Cancel the common factor of $2$ .

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Step 6.7.12.3.1

Factor $2$ out of $- 2$ .

$e^{2 x} v = 2 (- 1) \frac{x e^{2 x}}{2} - 2 (- \frac{e^{2 x}}{4}) + C$

Step 6.7.12.3.2

Cancel the common factor.

$e^{2 x} v = 2 \cdot - 1 \frac{x e^{2 x}}{2} - 2 (- \frac{e^{2 x}}{4}) + C$

Step 6.7.12.3.3

Rewrite the expression.

$e^{2 x} v = - (x e^{2 x}) - 2 (- \frac{e^{2 x}}{4}) + C$

Step 6.7.12.4

Cancel the common factor of $2$ .

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Step 6.7.12.4.1

Move the leading negative in $- \frac{e^{2 x}}{4}$ into the numerator.

$e^{2 x} v = - (x e^{2 x}) - 2 \frac{- e^{2 x}}{4} + C$

Step 6.7.12.4.2

Factor $2$ out of $- 2$ .

$e^{2 x} v = - (x e^{2 x}) + 2 (- 1) \frac{- e^{2 x}}{4} + C$

Step 6.7.12.4.3

Factor $2$ out of $4$ .

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

Step 6.7.12.4.4

Cancel the common factor.

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

Step 6.7.12.4.5

Rewrite the expression.

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

Step 6.7.12.5

Simplify each term.

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Step 6.7.12.5.1

Move the negative in front of the fraction.

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

Step 6.7.12.5.2

Multiply $- - \frac{e^{2 x}}{2}$ .

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Step 6.7.12.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.7.12.5.2.2

Multiply $\frac{e^{2 x}}{2}$ by $1$ .

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

Step 6.7.12.6

To write $- x e^{2 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.7.12.7

Combine $- x e^{2 x}$ and $\frac{2}{2}$ .

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

Step 6.7.12.8

Combine the numerators over the common denominator.

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

Step 6.7.12.9

Simplify the numerator.

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Step 6.7.12.9.1

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

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Step 6.7.12.9.1.1

Factor $e^{2 x}$ out of $- x e^{2 x} \cdot 2$ .

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

Step 6.7.12.9.1.2

Multiply by $1$ .

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

Step 6.7.12.9.1.3

Factor $e^{2 x}$ out of $e^{2 x} (- x \cdot 2) + e^{2 x} \cdot 1$ .

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

Step 6.7.12.9.2

Multiply $2$ by $- 1$ .

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

Step 6.7.12.10

Factor $- 1$ out of $- 2 x$ .

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

Step 6.7.12.11

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 6.7.12.12

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

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

Step 6.7.12.13

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

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

Step 6.7.12.14

Move the negative in front of the fraction.

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

Step 6.7.13

Remove parentheses.

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

Step 6.8

Solve for $v$ .

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Step 6.8.1

Combine $e^{2 x}$ and $\frac{1}{2}$ .

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

Step 6.8.2

Divide each term in $e^{2 x} v = - \frac{e^{2 x}}{2} \cdot (2 x - 1) + C$ by $e^{2 x}$ and simplify.

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Step 6.8.2.1

Divide each term in $e^{2 x} v = - \frac{e^{2 x}}{2} \cdot (2 x - 1) + C$ by $e^{2 x}$ .

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

Step 6.8.2.2

Simplify the left side.

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Step 6.8.2.2.1

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

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Step 6.8.2.2.1.1

Cancel the common factor.

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

Step 6.8.2.2.1.2

Divide $v$ by $1$ .

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

Step 6.8.2.3

Simplify the right side.

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Step 6.8.2.3.1

Combine the numerators over the common denominator.

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

Step 6.8.2.3.2

Simplify each term.

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Step 6.8.2.3.2.1

Apply the distributive property.

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

Step 6.8.2.3.2.2

Cancel the common factor of $2$ .

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Step 6.8.2.3.2.2.1

Move the leading negative in $- \frac{e^{2 x}}{2}$ into the numerator.

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

Step 6.8.2.3.2.2.2

Factor $2$ out of $2 x$ .

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

Step 6.8.2.3.2.2.3

Cancel the common factor.

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

Step 6.8.2.3.2.2.4

Rewrite the expression.

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

Step 6.8.2.3.2.3

Multiply $- \frac{e^{2 x}}{2} \cdot - 1$ .

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Step 6.8.2.3.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.8.2.3.2.3.2

Multiply $\frac{e^{2 x}}{2}$ by $1$ .

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

Step 6.8.2.3.2.4

Combine $- e^{2 x} x$ and $\frac{e^{2 x}}{2}$ using a common denominator.

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Step 6.8.2.3.2.4.1

Move $e^{2 x}$ .

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

Step 6.8.2.3.2.4.2

To write $- 1 x e^{2 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.8.2.3.2.4.3

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

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

Step 6.8.2.3.2.4.4

Combine the numerators over the common denominator.

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

Step 6.8.2.3.2.5

Simplify the numerator.

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Step 6.8.2.3.2.5.1

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

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Step 6.8.2.3.2.5.1.1

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

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

Step 6.8.2.3.2.5.1.2

Multiply by $1$ .

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

Step 6.8.2.3.2.5.1.3

Factor $e^{2 x}$ out of $e^{2 x} (- 1 x \cdot 2) + e^{2 x} \cdot 1$ .

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

Step 6.8.2.3.2.5.2

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

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

Step 6.8.2.3.2.5.3

Multiply $2$ by $- 1$ .

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

Step 6.8.2.3.3

To write $C$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.8.2.3.4

Simplify terms.

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Step 6.8.2.3.4.1

Combine $C$ and $\frac{2}{2}$ .

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

Step 6.8.2.3.4.2

Combine the numerators over the common denominator.

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

Step 6.8.2.3.5

Simplify the numerator.

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Step 6.8.2.3.5.1

Apply the distributive property.

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

Step 6.8.2.3.5.2

Rewrite using the commutative property of multiplication.

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

Step 6.8.2.3.5.3

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

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

Step 6.8.2.3.5.4

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

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

Step 6.8.2.3.6

Simplify with factoring out.

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Step 6.8.2.3.6.1

Factor $- 1$ out of $- 2 e^{2 x} x$ .

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

Step 6.8.2.3.6.2

Factor $- 1$ out of $e^{2 x}$ .

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

Step 6.8.2.3.6.3

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

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

Step 6.8.2.3.6.4

Factor $- 1$ out of $2 C$ .

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

Step 6.8.2.3.6.5

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

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

Step 6.8.2.3.6.6

Simplify the expression.

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Step 6.8.2.3.6.6.1

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

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

Step 6.8.2.3.6.6.2

Move the negative in front of the fraction.

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

Step 6.8.2.3.6.6.3

Reorder factors in $- \frac{2 e^{2 x} x - e^{2 x} - 2 C}{2}$ .

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

Step 6.8.2.3.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.8.2.3.8

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

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

Step 6.8.2.3.9

Move $2$ to the left of $e^{2 x}$ .

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

Step 7

Substitute $y^{- 2}$ for $v$ .

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