Calculus Examples

$\frac{d y}{d x} - 5 y = - \frac{5}{2} x y^{3}$

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 $\frac{d y}{d x} - 5 y = - \frac{5}{2} x y^{3}$ .

$- \frac{v'}{2 v^{\frac{3}{2}}} - 5 v^{- \frac{1}{2}} = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3}$

Step 6

Solve the substituted differential equation.

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Step 6.1

Multiply each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - 5 v^{- \frac{1}{2}} = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3}$ by $- (2 v^{\frac{3}{2}})$ to eliminate the fractions.

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Step 6.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - 5 v^{- \frac{1}{2}} = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3}$ by $- (2 v^{\frac{3}{2}})$ .

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} (- (2 v^{\frac{3}{2}})) - 5 v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

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.

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (- (2 v^{\frac{3}{2}})) - 5 v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.1.2

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

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (2 v^{\frac{3}{2}} (- 1)) - 5 v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (2 v^{\frac{3}{2}} \cdot - 1) - 5 v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 - 5 v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} - 5 v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.3

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

$\frac{d v}{d x} - 5 v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

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

$\frac{d v}{d x} - 5 (v^{\frac{3}{2}} v^{- \frac{1}{2}}) (- 1 \cdot (2)) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.2

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

$\frac{d v}{d x} - 5 v^{\frac{3}{2} - \frac{1}{2}} (- 1 \cdot (2)) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} - 5 v^{\frac{3 - 1}{2}} (- 1 \cdot (2)) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.4

Subtract $1$ from $3$ .

$\frac{d v}{d x} - 5 v^{\frac{2}{2}} (- 1 \cdot (2)) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.4.5

Divide $2$ by $2$ .

$\frac{d v}{d x} - 5 v^{1} (- 1 \cdot (2)) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.5

Simplify $- 5 v^{1} (- 1 \cdot (2))$ .

$\frac{d v}{d x} - 5 v (- 1 \cdot (2)) = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.6

Multiply $- 1$ by $2$ .

$\frac{d v}{d x} - 5 v \cdot - 2 = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.2.1.7

Multiply $- 2$ by $- 5$ .

$\frac{d v}{d x} + 10 v = - \frac{5}{2} x {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Combine $x$ and $\frac{5}{2}$ .

$\frac{d v}{d x} + 10 v = - \frac{x \cdot 5}{2} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.3.2

Simplify the expression.

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Step 6.1.3.2.1

Move $5$ to the left of $x$ .

$\frac{d v}{d x} + 10 v = - \frac{5 \cdot x}{2} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.3.2.2

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

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Step 6.1.3.2.2.1

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

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{- \frac{1}{2} \cdot 3} (- (2 v^{\frac{3}{2}}))$

Step 6.1.3.2.2.2

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

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Step 6.1.3.2.2.2.1

Multiply $3$ by $- 1$ .

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{- 3 (\frac{1}{2})} (- (2 v^{\frac{3}{2}}))$

Step 6.1.3.2.2.2.2

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

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{\frac{- 3}{2}} (- (2 v^{\frac{3}{2}}))$

Step 6.1.3.2.2.3

Move the negative in front of the fraction.

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{- \frac{3}{2}} (- (2 v^{\frac{3}{2}}))$

Step 6.1.3.3

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

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Step 6.1.3.3.1

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

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} (v^{\frac{3}{2}} v^{- \frac{3}{2}}) (- 1 \cdot (2))$

Step 6.1.3.3.2

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

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{\frac{3}{2} - \frac{3}{2}} (- 1 \cdot (2))$

Step 6.1.3.3.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{\frac{3 - 3}{2}} (- 1 \cdot (2))$

Step 6.1.3.3.4

Subtract $3$ from $3$ .

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{\frac{0}{2}} (- 1 \cdot (2))$

Step 6.1.3.3.5

Divide $0$ by $2$ .

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} v^{0} (- 1 \cdot (2))$

Step 6.1.3.4

Simplify $- \frac{5 x}{2} v^{0} (- 1 \cdot (2))$ .

$\frac{d v}{d x} + 10 v = - \frac{5 x}{2} (- 1 \cdot (2))$

Step 6.1.3.5

Cancel the common factor of $2$ .

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Step 6.1.3.5.1

Move the leading negative in $- \frac{5 x}{2}$ into the numerator.

$\frac{d v}{d x} + 10 v = \frac{- 5 x}{2} (- 1 \cdot (2))$

Step 6.1.3.5.2

Factor $2$ out of $- 1 \cdot (2)$ .

$\frac{d v}{d x} + 10 v = \frac{- 5 x}{2} (2 \cdot - 1)$

Step 6.1.3.5.3

Cancel the common factor.

$\frac{d v}{d x} + 10 v = \frac{- 5 x}{2} (2 \cdot - 1)$

Step 6.1.3.5.4

Rewrite the expression.

$\frac{d v}{d x} + 10 v = - 5 x \cdot - 1$

Step 6.1.3.6

Multiply $- 1$ by $- 5$ .

$\frac{d v}{d x} + 10 v = 5 x$

Step 6.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 10$ .

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Step 6.2.1

Set up the integration.

$e^{\int 10 d x}$

Step 6.2.2

Apply the constant rule.

$e^{10 x + C}$

Step 6.2.3

Remove the constant of integration.

$e^{10 x}$

Step 6.3

Multiply each term by the integrating factor $e^{10 x}$ .

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Step 6.3.1

Multiply each term by $e^{10 x}$ .

$e^{10 x} \frac{d v}{d x} + e^{10 x} (10 v) = e^{10 x} (5 x)$

Step 6.3.2

Rewrite using the commutative property of multiplication.

$e^{10 x} \frac{d v}{d x} + 10 e^{10 x} v = e^{10 x} (5 x)$

Step 6.3.3

Rewrite using the commutative property of multiplication.

$e^{10 x} \frac{d v}{d x} + 10 e^{10 x} v = 5 e^{10 x} x$

Step 6.3.4

Reorder factors in $e^{10 x} \frac{d v}{d x} + 10 e^{10 x} v = 5 e^{10 x} x$ .

$e^{10 x} \frac{d v}{d x} + 10 v e^{10 x} = 5 x e^{10 x}$

Step 6.4

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

$\frac{d}{d x} [e^{10 x} v] = 5 x e^{10 x}$

Step 6.5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{10 x} v] d x = \int 5 x e^{10 x} d x$

Step 6.6

Integrate the left side.

$e^{10 x} v = \int 5 x e^{10 x} d x$

Step 6.7

Integrate the right side.

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Step 6.7.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$e^{10 x} v = 5 \int x e^{10 x} d x$

Step 6.7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{10 x}$ .

$e^{10 x} v = 5 (x (\frac{1}{10} e^{10 x}) - \int \frac{1}{10} e^{10 x} d x)$

Step 6.7.3

Simplify.

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Step 6.7.3.1

Combine $\frac{1}{10}$ and $e^{10 x}$ .

$e^{10 x} v = 5 (x \frac{e^{10 x}}{10} - \int \frac{1}{10} e^{10 x} d x)$

Step 6.7.3.2

Combine $x$ and $\frac{e^{10 x}}{10}$ .

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \int \frac{1}{10} e^{10 x} d x)$

Step 6.7.3.3

Combine $\frac{1}{10}$ and $e^{10 x}$ .

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \int \frac{e^{10 x}}{10} d x)$

Step 6.7.4

Since $\frac{1}{10}$ is constant with respect to $x$ , move $\frac{1}{10}$ out of the integral.

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - (\frac{1}{10} \int e^{10 x} d x))$

Step 6.7.5

Let $u = 10 x$ . Then $d u = 10 d x$ , so $\frac{1}{10} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.7.5.1

Let $u = 10 x$ . Find $\frac{d u}{d x}$ .

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Step 6.7.5.1.1

Differentiate $10 x$ .

$\frac{d}{d x} [10 x]$

Step 6.7.5.1.2

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$10 \frac{d}{d x} [x]$

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 $n = 1$ .

$10 \cdot 1$

Step 6.7.5.1.4

Multiply $10$ by $1$ .

$10$

Step 6.7.5.2

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

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{1}{10} \int e^{u} \frac{1}{10} d u)$

Step 6.7.6

Combine $e^{u}$ and $\frac{1}{10}$ .

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{1}{10} \int \frac{e^{u}}{10} d u)$

Step 6.7.7

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{1}{10} (\frac{1}{10} \int e^{u} d u))$

Step 6.7.8

Simplify.

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Step 6.7.8.1

Multiply $\frac{1}{10}$ by $\frac{1}{10}$ .

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{1}{10 \cdot 10} \int e^{u} d u)$

Step 6.7.8.2

Multiply $10$ by $10$ .

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{1}{100} \int e^{u} d u)$

Step 6.7.9

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

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{1}{100} (e^{u} + C))$

Step 6.7.10

Rewrite $5 (\frac{x e^{10 x}}{10} - \frac{1}{100} (e^{u} + C))$ as $5 (\frac{1}{10} x e^{10 x} - \frac{1}{100} e^{u}) + C$ .

$e^{10 x} v = 5 (\frac{1}{10} x e^{10 x} - \frac{1}{100} e^{u}) + C$

Step 6.7.11

Replace all occurrences of $u$ with $10 x$ .

$e^{10 x} v = 5 (\frac{1}{10} x e^{10 x} - \frac{1}{100} e^{10 x}) + C$

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 $\frac{1}{10}$ and $x$ .

$e^{10 x} v = 5 (\frac{x}{10} e^{10 x} - \frac{1}{100} e^{10 x}) + C$

Step 6.7.12.1.2

Combine $\frac{x}{10}$ and $e^{10 x}$ .

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{1}{100} e^{10 x}) + C$

Step 6.7.12.1.3

Combine $e^{10 x}$ and $\frac{1}{100}$ .

$e^{10 x} v = 5 (\frac{x e^{10 x}}{10} - \frac{e^{10 x}}{100}) + C$

Step 6.7.12.2

Apply the distributive property.

$e^{10 x} v = 5 \frac{x e^{10 x}}{10} + 5 (- \frac{e^{10 x}}{100}) + C$

Step 6.7.12.3

Cancel the common factor of $5$ .

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Step 6.7.12.3.1

Factor $5$ out of $10$ .

$e^{10 x} v = 5 \frac{x e^{10 x}}{5 (2)} + 5 (- \frac{e^{10 x}}{100}) + C$

Step 6.7.12.3.2

Cancel the common factor.

$e^{10 x} v = 5 \frac{x e^{10 x}}{5 \cdot 2} + 5 (- \frac{e^{10 x}}{100}) + C$

Step 6.7.12.3.3

Rewrite the expression.

$e^{10 x} v = \frac{x e^{10 x}}{2} + 5 (- \frac{e^{10 x}}{100}) + C$

Step 6.7.12.4

Cancel the common factor of $5$ .

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Step 6.7.12.4.1

Move the leading negative in $- \frac{e^{10 x}}{100}$ into the numerator.

$e^{10 x} v = \frac{x e^{10 x}}{2} + 5 \frac{- e^{10 x}}{100} + C$

Step 6.7.12.4.2

Factor $5$ out of $100$ .

$e^{10 x} v = \frac{x e^{10 x}}{2} + 5 \frac{- e^{10 x}}{5 (20)} + C$

Step 6.7.12.4.3

Cancel the common factor.

$e^{10 x} v = \frac{x e^{10 x}}{2} + 5 \frac{- e^{10 x}}{5 \cdot 20} + C$

Step 6.7.12.4.4

Rewrite the expression.

$e^{10 x} v = \frac{x e^{10 x}}{2} + \frac{- e^{10 x}}{20} + C$

Step 6.7.12.5

Move the negative in front of the fraction.

$e^{10 x} v = \frac{x e^{10 x}}{2} - \frac{e^{10 x}}{20} + C$

Step 6.7.13

Reorder terms.

$e^{10 x} v = \frac{1}{2} x e^{10 x} - \frac{1}{20} e^{10 x} + C$

Step 6.8

Divide each term in $e^{10 x} v = \frac{1}{2} x e^{10 x} - \frac{1}{20} e^{10 x} + C$ by $e^{10 x}$ and simplify.

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Step 6.8.1

Divide each term in $e^{10 x} v = \frac{1}{2} x e^{10 x} - \frac{1}{20} e^{10 x} + C$ by $e^{10 x}$ .

$\frac{e^{10 x} v}{e^{10 x}} = \frac{\frac{1}{2} x e^{10 x}}{e^{10 x}} + \frac{- \frac{1}{20} e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}}$

Step 6.8.2

Simplify the left side.

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Step 6.8.2.1

Cancel the common factor of $e^{10 x}$ .

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Step 6.8.2.1.1

Cancel the common factor.

$\frac{e^{10 x} v}{e^{10 x}} = \frac{\frac{1}{2} x e^{10 x}}{e^{10 x}} + \frac{- \frac{1}{20} e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}}$

Step 6.8.2.1.2

Divide $v$ by $1$ .

$v = \frac{\frac{1}{2} x e^{10 x}}{e^{10 x}} + \frac{- \frac{1}{20} e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}}$

Step 6.8.3

Simplify the right side.

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Step 6.8.3.1

Simplify each term.

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Step 6.8.3.1.1

Cancel the common factor of $e^{10 x}$ .

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Step 6.8.3.1.1.1

Cancel the common factor.

$v = \frac{\frac{1}{2} x e^{10 x}}{e^{10 x}} + \frac{- \frac{1}{20} e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}}$

Step 6.8.3.1.1.2

Divide $\frac{1}{2} x$ by $1$ .

$v = \frac{1}{2} x + \frac{- \frac{1}{20} e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}}$

Step 6.8.3.1.2

Cancel the common factor of $e^{10 x}$ .

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Step 6.8.3.1.2.1

Cancel the common factor.

$v = \frac{1}{2} x + \frac{- \frac{1}{20} e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}}$

Step 6.8.3.1.2.2

Divide $- \frac{1}{20}$ by $1$ .

$v = \frac{1}{2} x - \frac{1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.2

Subtract $\frac{1}{20}$ from $\frac{1}{2} x$ .

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Step 6.8.3.2.1

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

$v = x \frac{1}{2} - \frac{1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.2.2

To write $x \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$v = x \frac{1}{2} \cdot \frac{20}{20} - \frac{1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.2.3

Combine $x \frac{1}{2}$ and $\frac{20}{20}$ .

$v = \frac{x \frac{1}{2} \cdot 20}{20} - \frac{1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.2.4

Combine the numerators over the common denominator.

$v = \frac{x \frac{1}{2} \cdot 20 - 1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.3

Simplify the numerator.

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Step 6.8.3.3.1

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

$v = \frac{\frac{x}{2} \cdot 20 - 1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.3.2

Cancel the common factor of $2$ .

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Step 6.8.3.3.2.1

Factor $2$ out of $20$ .

$v = \frac{\frac{x}{2} \cdot (2 (10)) - 1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.3.2.2

Cancel the common factor.

$v = \frac{\frac{x}{2} \cdot (2 \cdot 10) - 1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.3.2.3

Rewrite the expression.

$v = \frac{x \cdot 10 - 1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.3.3

Move $10$ to the left of $x$ .

$v = \frac{10 x - 1}{20} + \frac{C}{e^{10 x}}$

Step 6.8.3.4

To write $\frac{10 x - 1}{20}$ as a fraction with a common denominator, multiply by $\frac{e^{10 x}}{e^{10 x}}$ .

$v = \frac{10 x - 1}{20} \cdot \frac{e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}}$

Step 6.8.3.5

To write $\frac{C}{e^{10 x}}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$v = \frac{10 x - 1}{20} \cdot \frac{e^{10 x}}{e^{10 x}} + \frac{C}{e^{10 x}} \cdot \frac{20}{20}$

Step 6.8.3.6

Write each expression with a common denominator of $20 e^{10 x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.8.3.6.1

Multiply $\frac{10 x - 1}{20}$ by $\frac{e^{10 x}}{e^{10 x}}$ .

$v = \frac{(10 x - 1) e^{10 x}}{20 e^{10 x}} + \frac{C}{e^{10 x}} \cdot \frac{20}{20}$

Step 6.8.3.6.2

Multiply $\frac{C}{e^{10 x}}$ by $\frac{20}{20}$ .

$v = \frac{(10 x - 1) e^{10 x}}{20 e^{10 x}} + \frac{C \cdot 20}{e^{10 x} \cdot 20}$

Step 6.8.3.6.3

Reorder the factors of $e^{10 x} \cdot 20$ .

$v = \frac{(10 x - 1) e^{10 x}}{20 e^{10 x}} + \frac{C \cdot 20}{20 e^{10 x}}$

Step 6.8.3.7

Combine the numerators over the common denominator.

$v = \frac{(10 x - 1) e^{10 x} + C \cdot 20}{20 e^{10 x}}$

Step 6.8.3.8

Simplify the numerator.

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Step 6.8.3.8.1

Apply the distributive property.

$v = \frac{10 x e^{10 x} - 1 e^{10 x} + C \cdot 20}{20 e^{10 x}}$

Step 6.8.3.8.2

Rewrite $- 1 e^{10 x}$ as $- e^{10 x}$ .

$v = \frac{10 x e^{10 x} - e^{10 x} + C \cdot 20}{20 e^{10 x}}$

Step 6.8.3.8.3

Move $20$ to the left of $C$ .

$v = \frac{10 x e^{10 x} - e^{10 x} + 20 C}{20 e^{10 x}}$

Step 7

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

$y^{- 2} = \frac{10 x e^{10 x} - e^{10 x} + 20 C}{20 e^{10 x}}$