Calculus Examples

$\frac{d y}{d x} - y = x y^{5}$

Step 1

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{5}$ .

$v = y^{- 4}$

Step 2

Solve the equation for $y$ .

$y = v^{- \frac{1}{4}}$

Step 3

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

$y' = v^{- \frac{1}{4}}$

Step 4

Take the derivative of $v^{- \frac{1}{4}}$ with respect to $x$ .

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Step 4.1

Take the derivative of $v^{- \frac{1}{4}}$ .

$y' = \frac{d}{d x} [v^{- \frac{1}{4}}]$

Step 4.2

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

$y' = \frac{d}{d x} [\frac{1}{v^{\frac{1}{4}}}]$

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 $g (x) = v^{\frac{1}{4}}$ .

$y' = \frac{v^{\frac{1}{4}} \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v^{\frac{1}{4}}]}{{(v^{\frac{1}{4}})}^{2}}$

Step 4.4

Differentiate using the Constant Rule.

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Step 4.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v^{\frac{1}{4}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{4}}]}{{(v^{\frac{1}{4}})}^{2}}$

Step 4.4.2

Multiply the exponents in ${(v^{\frac{1}{4}})}^{2}$ .

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Step 4.4.2.1

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

$y' = \frac{v^{\frac{1}{4}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{4} \cdot 2}}$

Step 4.4.2.2

Cancel the common factor of $2$ .

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Step 4.4.2.2.1

Factor $2$ out of $4$ .

$y' = \frac{v^{\frac{1}{4}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{2 (2)} \cdot 2}}$

Step 4.4.2.2.2

Cancel the common factor.

$y' = \frac{v^{\frac{1}{4}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{2 \cdot 2} \cdot 2}}$

Step 4.4.2.2.3

Rewrite the expression.

$y' = \frac{v^{\frac{1}{4}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{2}}}$

Step 4.4.3

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

$y' = \frac{v^{\frac{1}{4}} \cdot 0 - \frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{2}}}$

Step 4.4.4

Simplify the expression.

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Step 4.4.4.1

Multiply $v^{\frac{1}{4}}$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{2}}}$

Step 4.4.4.2

Subtract $\frac{d}{d x} [v^{\frac{1}{4}}]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{2}}}$

Step 4.4.4.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v^{\frac{1}{4}}]}{v^{\frac{1}{2}}}$

Step 4.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{4}}$ and $g (x) = v$ .

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Step 4.5.1

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

$y' = - \frac{\frac{d}{d u} [u^{\frac{1}{4}}] \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{4}$ .

$y' = - \frac{\frac{1}{4} u^{\frac{1}{4} - 1} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.5.3

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

$y' = - \frac{\frac{1}{4} v^{\frac{1}{4} - 1} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y' = - \frac{\frac{1}{4} v^{\frac{1}{4} - 1 \cdot \frac{4}{4}} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.7

Combine $- 1$ and $\frac{4}{4}$ .

$y' = - \frac{\frac{1}{4} v^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.8

Combine the numerators over the common denominator.

$y' = - \frac{\frac{1}{4} v^{\frac{1 - 1 \cdot 4}{4}} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Multiply $- 1$ by $4$ .

$y' = - \frac{\frac{1}{4} v^{\frac{1 - 4}{4}} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.9.2

Subtract $4$ from $1$ .

$y' = - \frac{\frac{1}{4} v^{\frac{- 3}{4}} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.10

Move the negative in front of the fraction.

$y' = - \frac{\frac{1}{4} v^{- \frac{3}{4}} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.11

Combine $\frac{1}{4}$ and $v^{- \frac{3}{4}}$ .

$y' = - \frac{\frac{v^{- \frac{3}{4}}}{4} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.12

Move $v^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y' = - \frac{\frac{1}{4 v^{\frac{3}{4}}} \frac{d}{d x} [v]}{v^{\frac{1}{2}}}$

Step 4.13

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

$y' = - \frac{\frac{1}{4 v^{\frac{3}{4}}} v'}{v^{\frac{1}{2}}}$

Step 4.14

Combine $\frac{1}{4 v^{\frac{3}{4}}}$ and $v'$ .

$y' = - \frac{\frac{v'}{4 v^{\frac{3}{4}}}}{v^{\frac{1}{2}}}$

Step 4.15

Rewrite $\frac{\frac{v'}{4 v^{\frac{3}{4}}}}{v^{\frac{1}{2}}}$ as a product.

$y' = - (\frac{v'}{4 v^{\frac{3}{4}}} \cdot \frac{1}{v^{\frac{1}{2}}})$

Step 4.16

Multiply $\frac{v'}{4 v^{\frac{3}{4}}}$ by $\frac{1}{v^{\frac{1}{2}}}$ .

$y' = - \frac{v'}{4 v^{\frac{3}{4}} v^{\frac{1}{2}}}$

Step 4.17

Multiply $v^{\frac{3}{4}}$ by $v^{\frac{1}{2}}$ by adding the exponents.

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Step 4.17.1

Move $v^{\frac{1}{2}}$ .

$y' = - \frac{v'}{4 (v^{\frac{1}{2}} v^{\frac{3}{4}})}$

Step 4.17.2

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

$y' = - \frac{v'}{4 v^{\frac{1}{2} + \frac{3}{4}}}$

Step 4.17.3

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

$y' = - \frac{v'}{4 v^{\frac{1}{2} \cdot \frac{2}{2} + \frac{3}{4}}}$

Step 4.17.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.17.4.1

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

$y' = - \frac{v'}{4 v^{\frac{2}{2 \cdot 2} + \frac{3}{4}}}$

Step 4.17.4.2

Multiply $2$ by $2$ .

$y' = - \frac{v'}{4 v^{\frac{2}{4} + \frac{3}{4}}}$

Step 4.17.5

Combine the numerators over the common denominator.

$y' = - \frac{v'}{4 v^{\frac{2 + 3}{4}}}$

Step 4.17.6

Add $2$ and $3$ .

$y' = - \frac{v'}{4 v^{\frac{5}{4}}}$

Step 5

Substitute $- \frac{v'}{4 v^{\frac{5}{4}}}$ for $\frac{d y}{d x}$ and $v^{- \frac{1}{4}}$ for $y$ in the original equation $\frac{d y}{d x} - y = x y^{5}$ .

$- \frac{v'}{4 v^{\frac{5}{4}}} - v^{- \frac{1}{4}} = x {(v^{- \frac{1}{4}})}^{5}$

Step 6

Solve the substituted differential equation.

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Step 6.1

Multiply each term in $- \frac{\frac{d v}{d x}}{4 v^{\frac{5}{4}}} - v^{- \frac{1}{4}} = x {(v^{- \frac{1}{4}})}^{5}$ by $- (4 v^{\frac{5}{4}})$ to eliminate the fractions.

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Step 6.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{4 v^{\frac{5}{4}}} - v^{- \frac{1}{4}} = x {(v^{- \frac{1}{4}})}^{5}$ by $- (4 v^{\frac{5}{4}})$ .

$- \frac{\frac{d v}{d x}}{4 v^{\frac{5}{4}}} (- (4 v^{\frac{5}{4}})) - v^{- \frac{1}{4}} (- (4 v^{\frac{5}{4}})) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

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 $4 v^{\frac{5}{4}}$ .

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Step 6.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{4 v^{\frac{5}{4}}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{4 v^{\frac{5}{4}}} (- (4 v^{\frac{5}{4}})) - v^{- \frac{1}{4}} (- (4 v^{\frac{5}{4}})) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.1.2

Factor $4 v^{\frac{5}{4}}$ out of $- (4 v^{\frac{5}{4}})$ .

$\frac{- \frac{d v}{d x}}{4 v^{\frac{5}{4}}} (4 v^{\frac{5}{4}} (- 1)) - v^{- \frac{1}{4}} (- (4 v^{\frac{5}{4}})) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{4 v^{\frac{5}{4}}} (4 v^{\frac{5}{4}} \cdot - 1) - v^{- \frac{1}{4}} (- (4 v^{\frac{5}{4}})) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 - v^{- \frac{1}{4}} (- (4 v^{\frac{5}{4}})) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} - v^{- \frac{1}{4}} (- (4 v^{\frac{5}{4}})) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.3

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

$\frac{d v}{d x} - v^{- \frac{1}{4}} (- (4 v^{\frac{5}{4}})) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.4

Multiply $v^{- \frac{1}{4}}$ by $v^{\frac{5}{4}}$ by adding the exponents.

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Step 6.1.2.1.4.1

Move $v^{\frac{5}{4}}$ .

$\frac{d v}{d x} - (v^{\frac{5}{4}} v^{- \frac{1}{4}}) (- 1 \cdot (4)) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.4.2

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

$\frac{d v}{d x} - v^{\frac{5}{4} - \frac{1}{4}} (- 1 \cdot (4)) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.4.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} - v^{\frac{5 - 1}{4}} (- 1 \cdot (4)) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.4.4

Subtract $1$ from $5$ .

$\frac{d v}{d x} - v^{\frac{4}{4}} (- 1 \cdot (4)) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.4.5

Divide $4$ by $4$ .

$\frac{d v}{d x} - v^{1} (- 1 \cdot (4)) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.5

Simplify $- v^{1} (- 1 \cdot (4))$ .

$\frac{d v}{d x} - v (- 1 \cdot (4)) = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.6

Multiply $- 1$ by $4$ .

$\frac{d v}{d x} - v \cdot - 4 = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.2.1.7

Multiply $- 4$ by $- 1$ .

$\frac{d v}{d x} + 4 v = x {(v^{- \frac{1}{4}})}^{5} (- (4 v^{\frac{5}{4}}))$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} + 4 v = - 1 \cdot 4 x {(v^{- \frac{1}{4}})}^{5} v^{\frac{5}{4}}$

Step 6.1.3.2

Multiply $- 1$ by $4$ .

$\frac{d v}{d x} + 4 v = - 4 x {(v^{- \frac{1}{4}})}^{5} v^{\frac{5}{4}}$

Step 6.1.3.3

Multiply the exponents in ${(v^{- \frac{1}{4}})}^{5}$ .

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Step 6.1.3.3.1

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

$\frac{d v}{d x} + 4 v = - 4 x v^{- \frac{1}{4} \cdot 5} v^{\frac{5}{4}}$

Step 6.1.3.3.2

Multiply $- \frac{1}{4} \cdot 5$ .

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Step 6.1.3.3.2.1

Multiply $5$ by $- 1$ .

$\frac{d v}{d x} + 4 v = - 4 x v^{- 5 (\frac{1}{4})} v^{\frac{5}{4}}$

Step 6.1.3.3.2.2

Combine $- 5$ and $\frac{1}{4}$ .

$\frac{d v}{d x} + 4 v = - 4 x v^{\frac{- 5}{4}} v^{\frac{5}{4}}$

Step 6.1.3.3.3

Move the negative in front of the fraction.

$\frac{d v}{d x} + 4 v = - 4 x v^{- \frac{5}{4}} v^{\frac{5}{4}}$

Step 6.1.3.4

Multiply $v^{- \frac{5}{4}}$ by $v^{\frac{5}{4}}$ by adding the exponents.

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Step 6.1.3.4.1

Move $v^{\frac{5}{4}}$ .

$\frac{d v}{d x} + 4 v = - 4 x (v^{\frac{5}{4}} v^{- \frac{5}{4}})$

Step 6.1.3.4.2

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

$\frac{d v}{d x} + 4 v = - 4 x v^{\frac{5}{4} - \frac{5}{4}}$

Step 6.1.3.4.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} + 4 v = - 4 x v^{\frac{5 - 5}{4}}$

Step 6.1.3.4.4

Subtract $5$ from $5$ .

$\frac{d v}{d x} + 4 v = - 4 x v^{\frac{0}{4}}$

Step 6.1.3.4.5

Divide $0$ by $4$ .

$\frac{d v}{d x} + 4 v = - 4 x v^{0}$

Step 6.1.3.5

Simplify $- 4 x v^{0}$ .

$\frac{d v}{d x} + 4 v = - 4 x$

Step 6.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 4$ .

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Step 6.2.1

Set up the integration.

$e^{\int 4 d x}$

Step 6.2.2

Apply the constant rule.

$e^{4 x + C}$

Step 6.2.3

Remove the constant of integration.

$e^{4 x}$

Step 6.3

Multiply each term by the integrating factor $e^{4 x}$ .

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Step 6.3.1

Multiply each term by $e^{4 x}$ .

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

Step 6.3.2

Rewrite using the commutative property of multiplication.

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

Step 6.3.3

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d v}{d x} + 4 e^{4 x} v = - 4 e^{4 x} x$

Step 6.3.4

Reorder factors in $e^{4 x} \frac{d v}{d x} + 4 e^{4 x} v = - 4 e^{4 x} x$ .

$e^{4 x} \frac{d v}{d x} + 4 v e^{4 x} = - 4 x e^{4 x}$

Step 6.4

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

$\frac{d}{d x} [e^{4 x} v] = - 4 x e^{4 x}$

Step 6.5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{4 x} v] d x = \int - 4 x e^{4 x} d x$

Step 6.6

Integrate the left side.

$e^{4 x} v = \int - 4 x e^{4 x} d x$

Step 6.7

Integrate the right side.

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Step 6.7.1

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$e^{4 x} v = - 4 \int x e^{4 x} d x$

Step 6.7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{4 x}$ .

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

Step 6.7.3

Simplify.

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Step 6.7.3.1

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

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

Step 6.7.3.2

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

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

Step 6.7.3.3

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

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

Step 6.7.4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 6.7.5

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.7.5.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 6.7.5.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 6.7.5.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

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 $n = 1$ .

$4 \cdot 1$

Step 6.7.5.1.4

Multiply $4$ by $1$ .

$4$

Step 6.7.5.2

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

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

Step 6.7.6

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

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

Step 6.7.7

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

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

Step 6.7.8

Simplify.

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Step 6.7.8.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

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

Step 6.7.8.2

Multiply $4$ by $4$ .

$e^{4 x} v = - 4 (\frac{x e^{4 x}}{4} - \frac{1}{16} \int e^{u} d u)$

Step 6.7.9

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

$e^{4 x} v = - 4 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C))$

Step 6.7.10

Rewrite $- 4 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C))$ as $- 4 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u}) + C$ .

$e^{4 x} v = - 4 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u}) + C$

Step 6.7.11

Replace all occurrences of $u$ with $4 x$ .

$e^{4 x} v = - 4 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{4 x}) + C$

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 $\frac{1}{4}$ and $x$ .

$e^{4 x} v = - 4 (\frac{x}{4} e^{4 x} - \frac{1}{16} e^{4 x}) + C$

Step 6.7.12.1.2

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

$e^{4 x} v = - 4 (\frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x}) + C$

Step 6.7.12.1.3

Combine $e^{4 x}$ and $\frac{1}{16}$ .

$e^{4 x} v = - 4 (\frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}) + C$

Step 6.7.12.2

Apply the distributive property.

$e^{4 x} v = - 4 \frac{x e^{4 x}}{4} - 4 (- \frac{e^{4 x}}{16}) + C$

Step 6.7.12.3

Cancel the common factor of $4$ .

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Step 6.7.12.3.1

Factor $4$ out of $- 4$ .

$e^{4 x} v = 4 (- 1) \frac{x e^{4 x}}{4} - 4 (- \frac{e^{4 x}}{16}) + C$

Step 6.7.12.3.2

Cancel the common factor.

$e^{4 x} v = 4 \cdot - 1 \frac{x e^{4 x}}{4} - 4 (- \frac{e^{4 x}}{16}) + C$

Step 6.7.12.3.3

Rewrite the expression.

$e^{4 x} v = - (x e^{4 x}) - 4 (- \frac{e^{4 x}}{16}) + C$

Step 6.7.12.4

Cancel the common factor of $4$ .

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Step 6.7.12.4.1

Move the leading negative in $- \frac{e^{4 x}}{16}$ into the numerator.

$e^{4 x} v = - (x e^{4 x}) - 4 \frac{- e^{4 x}}{16} + C$

Step 6.7.12.4.2

Factor $4$ out of $- 4$ .

$e^{4 x} v = - (x e^{4 x}) + 4 (- 1) \frac{- e^{4 x}}{16} + C$

Step 6.7.12.4.3

Factor $4$ out of $16$ .

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

Step 6.7.12.4.4

Cancel the common factor.

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

Step 6.7.12.4.5

Rewrite the expression.

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

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.

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

Step 6.7.12.5.2

Multiply $- - \frac{e^{4 x}}{4}$ .

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Step 6.7.12.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.7.12.5.2.2

Multiply $\frac{e^{4 x}}{4}$ by $1$ .

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

Step 6.7.12.6

To write $- x e^{4 x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e^{4 x} v = - x e^{4 x} \cdot \frac{4}{4} + \frac{e^{4 x}}{4} + C$

Step 6.7.12.7

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

$e^{4 x} v = \frac{- x e^{4 x} \cdot 4}{4} + \frac{e^{4 x}}{4} + C$

Step 6.7.12.8

Combine the numerators over the common denominator.

$e^{4 x} v = \frac{- x e^{4 x} \cdot 4 + e^{4 x}}{4} + C$

Step 6.7.12.9

Simplify the numerator.

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Step 6.7.12.9.1

Factor $e^{4 x}$ out of $- x e^{4 x} \cdot 4 + e^{4 x}$ .

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Step 6.7.12.9.1.1

Factor $e^{4 x}$ out of $- x e^{4 x} \cdot 4$ .

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

Step 6.7.12.9.1.2

Multiply by $1$ .

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

Step 6.7.12.9.1.3

Factor $e^{4 x}$ out of $e^{4 x} (- x \cdot 4) + e^{4 x} \cdot 1$ .

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

Step 6.7.12.9.2

Multiply $4$ by $- 1$ .

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

Step 6.7.12.10

Factor $- 1$ out of $- 4 x$ .

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

Step 6.7.12.11

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

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

Step 6.7.12.12

Factor $- 1$ out of $- (4 x) - 1 (- 1)$ .

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

Step 6.7.12.13

Rewrite $- (4 x - 1)$ as $- 1 (4 x - 1)$ .

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

Step 6.7.12.14

Move the negative in front of the fraction.

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

Step 6.7.13

Remove parentheses.

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

Step 6.8

Solve for $v$ .

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Step 6.8.1

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

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

Step 6.8.2

Divide each term in $e^{4 x} v = - \frac{e^{4 x}}{4} \cdot (4 x - 1) + C$ by $e^{4 x}$ and simplify.

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Step 6.8.2.1

Divide each term in $e^{4 x} v = - \frac{e^{4 x}}{4} \cdot (4 x - 1) + C$ by $e^{4 x}$ .

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

Step 6.8.2.2

Simplify the left side.

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Step 6.8.2.2.1

Cancel the common factor of $e^{4 x}$ .

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Step 6.8.2.2.1.1

Cancel the common factor.

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

Step 6.8.2.2.1.2

Divide $v$ by $1$ .

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

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.

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

Step 6.8.2.3.2

Simplify each term.

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Step 6.8.2.3.2.1

Apply the distributive property.

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

Step 6.8.2.3.2.2

Cancel the common factor of $4$ .

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Step 6.8.2.3.2.2.1

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

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

Step 6.8.2.3.2.2.2

Factor $4$ out of $4 x$ .

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

Step 6.8.2.3.2.2.3

Cancel the common factor.

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

Step 6.8.2.3.2.2.4

Rewrite the expression.

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

Step 6.8.2.3.2.3

Multiply $- \frac{e^{4 x}}{4} \cdot - 1$ .

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Step 6.8.2.3.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.8.2.3.2.3.2

Multiply $\frac{e^{4 x}}{4}$ by $1$ .

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

Step 6.8.2.3.2.4

Combine $- e^{4 x} x$ and $\frac{e^{4 x}}{4}$ using a common denominator.

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Step 6.8.2.3.2.4.1

Move $e^{4 x}$ .

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

Step 6.8.2.3.2.4.2

To write $- 1 x e^{4 x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 6.8.2.3.2.4.3

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

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

Step 6.8.2.3.2.4.4

Combine the numerators over the common denominator.

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

Step 6.8.2.3.2.5

Simplify the numerator.

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Step 6.8.2.3.2.5.1

Factor $e^{4 x}$ out of $- 1 x e^{4 x} \cdot 4 + e^{4 x}$ .

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Step 6.8.2.3.2.5.1.1

Factor $e^{4 x}$ out of $- 1 x e^{4 x} \cdot 4$ .

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

Step 6.8.2.3.2.5.1.2

Multiply by $1$ .

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

Step 6.8.2.3.2.5.1.3

Factor $e^{4 x}$ out of $e^{4 x} (- 1 x \cdot 4) + e^{4 x} \cdot 1$ .

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

Step 6.8.2.3.2.5.2

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

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

Step 6.8.2.3.2.5.3

Multiply $4$ by $- 1$ .

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

Step 6.8.2.3.3

To write $C$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 6.8.2.3.4

Simplify terms.

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Step 6.8.2.3.4.1

Combine $C$ and $\frac{4}{4}$ .

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

Step 6.8.2.3.4.2

Combine the numerators over the common denominator.

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

Step 6.8.2.3.5

Simplify the numerator.

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Step 6.8.2.3.5.1

Apply the distributive property.

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

Step 6.8.2.3.5.2

Rewrite using the commutative property of multiplication.

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

Step 6.8.2.3.5.3

Multiply $e^{4 x}$ by $1$ .

$v = \frac{\frac{- 4 e^{4 x} x + e^{4 x} + C \cdot 4}{4}}{e^{4 x}}$

Step 6.8.2.3.5.4

Move $4$ to the left of $C$ .

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

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 $- 4 e^{4 x} x$ .

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

Step 6.8.2.3.6.2

Factor $- 1$ out of $e^{4 x}$ .

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

Step 6.8.2.3.6.3

Factor $- 1$ out of $- (4 e^{4 x} x) - 1 (- e^{4 x})$ .

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

Step 6.8.2.3.6.4

Factor $- 1$ out of $4 C$ .

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

Step 6.8.2.3.6.5

Factor $- 1$ out of $- (4 e^{4 x} x - e^{4 x}) - (- 4 C)$ .

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

Step 6.8.2.3.6.6

Simplify the expression.

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Step 6.8.2.3.6.6.1

Rewrite $- (4 e^{4 x} x - e^{4 x} - 4 C)$ as $- 1 (4 e^{4 x} x - e^{4 x} - 4 C)$ .

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

Step 6.8.2.3.6.6.2

Move the negative in front of the fraction.

$v = \frac{- \frac{4 e^{4 x} x - e^{4 x} - 4 C}{4}}{e^{4 x}}$

Step 6.8.2.3.6.6.3

Reorder factors in $- \frac{4 e^{4 x} x - e^{4 x} - 4 C}{4}$ .

$v = \frac{- \frac{4 x e^{4 x} - e^{4 x} - 4 C}{4}}{e^{4 x}}$

Step 6.8.2.3.7

Multiply the numerator by the reciprocal of the denominator.

$v = - \frac{4 x e^{4 x} - e^{4 x} - 4 C}{4} \cdot \frac{1}{e^{4 x}}$

Step 6.8.2.3.8

Multiply $\frac{1}{e^{4 x}}$ by $\frac{4 x e^{4 x} - e^{4 x} - 4 C}{4}$ .

$v = - \frac{4 x e^{4 x} - e^{4 x} - 4 C}{e^{4 x} \cdot 4}$

Step 6.8.2.3.9

Move $4$ to the left of $e^{4 x}$ .

$v = - \frac{4 x e^{4 x} - e^{4 x} - 4 C}{4 e^{4 x}}$

Step 7

Substitute $y^{- 4}$ for $v$ .

$y^{- 4} = - \frac{4 x e^{4 x} - e^{4 x} - 4 C}{4 e^{4 x}}$