Calculus Examples

$\frac{d y}{d x} + \frac{y}{x} = x^{2} y^{6}$

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

Tap for more steps...

Step 1.1

Factor $y$ out of $\frac{y}{x}$ .

$\frac{d y}{d x} + y \frac{1}{x} = x^{2} y^{6}$

Step 1.2

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

$\frac{d y}{d x} + \frac{1}{x} y = x^{2} y^{6}$

Step 2

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{6}$ .

$v = y^{- 5}$

Step 3

Solve the equation for $y$ .

$y = v^{- \frac{1}{5}}$

Step 4

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

$y' = v^{- \frac{1}{5}}$

Step 5

Take the derivative of $v^{- \frac{1}{5}}$ with respect to $x$ .

Tap for more steps...

Step 5.1

Take the derivative of $v^{- \frac{1}{5}}$ .

$y' = \frac{d}{d x} [v^{- \frac{1}{5}}]$

Step 5.2

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

$y' = \frac{d}{d x} [\frac{1}{v^{\frac{1}{5}}}]$

Step 5.3

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 $g (x) = v^{\frac{1}{5}}$ .

$y' = \frac{v^{\frac{1}{5}} \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v^{\frac{1}{5}}]}{{(v^{\frac{1}{5}})}^{2}}$

Step 5.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 5.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v^{\frac{1}{5}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{5}}]}{{(v^{\frac{1}{5}})}^{2}}$

Step 5.4.2

Multiply the exponents in ${(v^{\frac{1}{5}})}^{2}$ .

Tap for more steps...

Step 5.4.2.1

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

$y' = \frac{v^{\frac{1}{5}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{5}}]}{v^{\frac{1}{5} \cdot 2}}$

Step 5.4.2.2

Combine $\frac{1}{5}$ and $2$ .

$y' = \frac{v^{\frac{1}{5}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{5}}]}{v^{\frac{2}{5}}}$

Step 5.4.3

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

$y' = \frac{v^{\frac{1}{5}} \cdot 0 - \frac{d}{d x} [v^{\frac{1}{5}}]}{v^{\frac{2}{5}}}$

Step 5.4.4

Simplify the expression.

Tap for more steps...

Step 5.4.4.1

Multiply $v^{\frac{1}{5}}$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v^{\frac{1}{5}}]}{v^{\frac{2}{5}}}$

Step 5.4.4.2

Subtract $\frac{d}{d x} [v^{\frac{1}{5}}]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v^{\frac{1}{5}}]}{v^{\frac{2}{5}}}$

Step 5.4.4.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v^{\frac{1}{5}}]}{v^{\frac{2}{5}}}$

Step 5.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = v$ .

Tap for more steps...

Step 5.5.1

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

$y' = - \frac{\frac{d}{d u} [u^{\frac{1}{5}}] \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{5}$ .

$y' = - \frac{\frac{1}{5} u^{\frac{1}{5} - 1} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.5.3

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

$y' = - \frac{\frac{1}{5} v^{\frac{1}{5} - 1} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y' = - \frac{\frac{1}{5} v^{\frac{1}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.7

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

$y' = - \frac{\frac{1}{5} v^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.8

Combine the numerators over the common denominator.

$y' = - \frac{\frac{1}{5} v^{\frac{1 - 1 \cdot 5}{5}} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.9

Simplify the numerator.

Tap for more steps...

Step 5.9.1

Multiply $- 1$ by $5$ .

$y' = - \frac{\frac{1}{5} v^{\frac{1 - 5}{5}} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.9.2

Subtract $5$ from $1$ .

$y' = - \frac{\frac{1}{5} v^{\frac{- 4}{5}} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.10

Move the negative in front of the fraction.

$y' = - \frac{\frac{1}{5} v^{- \frac{4}{5}} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.11

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

$y' = - \frac{\frac{v^{- \frac{4}{5}}}{5} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.12

Move $v^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y' = - \frac{\frac{1}{5 v^{\frac{4}{5}}} \frac{d}{d x} [v]}{v^{\frac{2}{5}}}$

Step 5.13

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

$y' = - \frac{\frac{1}{5 v^{\frac{4}{5}}} v'}{v^{\frac{2}{5}}}$

Step 5.14

Combine $\frac{1}{5 v^{\frac{4}{5}}}$ and $v'$ .

$y' = - \frac{\frac{v'}{5 v^{\frac{4}{5}}}}{v^{\frac{2}{5}}}$

Step 5.15

Rewrite $\frac{\frac{v'}{5 v^{\frac{4}{5}}}}{v^{\frac{2}{5}}}$ as a product.

$y' = - (\frac{v'}{5 v^{\frac{4}{5}}} \cdot \frac{1}{v^{\frac{2}{5}}})$

Step 5.16

Multiply $\frac{v'}{5 v^{\frac{4}{5}}}$ by $\frac{1}{v^{\frac{2}{5}}}$ .

$y' = - \frac{v'}{5 v^{\frac{4}{5}} v^{\frac{2}{5}}}$

Step 5.17

Multiply $v^{\frac{4}{5}}$ by $v^{\frac{2}{5}}$ by adding the exponents.

Tap for more steps...

Step 5.17.1

Move $v^{\frac{2}{5}}$ .

$y' = - \frac{v'}{5 (v^{\frac{2}{5}} v^{\frac{4}{5}})}$

Step 5.17.2

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

$y' = - \frac{v'}{5 v^{\frac{2}{5} + \frac{4}{5}}}$

Step 5.17.3

Combine the numerators over the common denominator.

$y' = - \frac{v'}{5 v^{\frac{2 + 4}{5}}}$

Step 5.17.4

Add $2$ and $4$ .

$y' = - \frac{v'}{5 v^{\frac{6}{5}}}$

Step 6

Substitute $- \frac{v'}{5 v^{\frac{6}{5}}}$ for $\frac{d y}{d x}$ and $v^{- \frac{1}{5}}$ for $y$ in the original equation $\frac{d y}{d x} + \frac{1}{x} y = x^{2} y^{6}$ .

$- \frac{v'}{5 v^{\frac{6}{5}}} + \frac{1}{x} v^{- \frac{1}{5}} = x^{2} {(v^{- \frac{1}{5}})}^{6}$

Step 7

Solve the substituted differential equation.

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{5 v^{\frac{6}{5}}} + \frac{1}{x} v^{- \frac{1}{5}} = x^{2} {(v^{- \frac{1}{5}})}^{6}$ by $- (5 v^{\frac{6}{5}})$ to eliminate the fractions.

Tap for more steps...

Step 7.1.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{5 v^{\frac{6}{5}}} + \frac{1}{x} v^{- \frac{1}{5}} = x^{2} {(v^{- \frac{1}{5}})}^{6}$ by $- (5 v^{\frac{6}{5}})$ .

$- \frac{\frac{d v}{d x}}{5 v^{\frac{6}{5}}} (- (5 v^{\frac{6}{5}})) + \frac{1}{x} v^{- \frac{1}{5}} (- (5 v^{\frac{6}{5}})) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2

Simplify the left side.

Tap for more steps...

Step 7.1.1.2.1

Simplify each term.

Tap for more steps...

Step 7.1.1.2.1.1

Cancel the common factor of $5 v^{\frac{6}{5}}$ .

Tap for more steps...

Step 7.1.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{5 v^{\frac{6}{5}}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{5 v^{\frac{6}{5}}} (- (5 v^{\frac{6}{5}})) + \frac{1}{x} v^{- \frac{1}{5}} (- (5 v^{\frac{6}{5}})) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.1.2

Factor $5 v^{\frac{6}{5}}$ out of $- (5 v^{\frac{6}{5}})$ .

$\frac{- \frac{d v}{d x}}{5 v^{\frac{6}{5}}} (5 v^{\frac{6}{5}} (- 1)) + \frac{1}{x} v^{- \frac{1}{5}} (- (5 v^{\frac{6}{5}})) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{5 v^{\frac{6}{5}}} (5 v^{\frac{6}{5}} \cdot - 1) + \frac{1}{x} v^{- \frac{1}{5}} (- (5 v^{\frac{6}{5}})) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 + \frac{1}{x} v^{- \frac{1}{5}} (- (5 v^{\frac{6}{5}})) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} + \frac{1}{x} v^{- \frac{1}{5}} (- (5 v^{\frac{6}{5}})) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.3

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

$\frac{d v}{d x} + \frac{1}{x} v^{- \frac{1}{5}} (- (5 v^{\frac{6}{5}})) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - 1 \cdot 5 \frac{1}{x} v^{- \frac{1}{5}} v^{\frac{6}{5}} = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.5

Multiply $v^{- \frac{1}{5}}$ by $v^{\frac{6}{5}}$ by adding the exponents.

Tap for more steps...

Step 7.1.1.2.1.5.1

Move $v^{\frac{6}{5}}$ .

$\frac{d v}{d x} - 1 \cdot 5 \frac{1}{x} (v^{\frac{6}{5}} v^{- \frac{1}{5}}) = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.5.2

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

$\frac{d v}{d x} - 1 \cdot 5 \frac{1}{x} v^{\frac{6}{5} - \frac{1}{5}} = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.5.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} - 1 \cdot 5 \frac{1}{x} v^{\frac{6 - 1}{5}} = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.5.4

Subtract $1$ from $6$ .

$\frac{d v}{d x} - 1 \cdot 5 \frac{1}{x} v^{\frac{5}{5}} = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.5.5

Divide $5$ by $5$ .

$\frac{d v}{d x} - 1 \cdot 5 \frac{1}{x} v^{1} = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.6

Simplify $- 1 \cdot 5 \frac{1}{x} v^{1}$ .

$\frac{d v}{d x} - 1 \cdot 5 \frac{1}{x} v = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.7

Multiply $- 1$ by $5$ .

$\frac{d v}{d x} - 5 \frac{1}{x} v = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.8

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

$\frac{d v}{d x} + \frac{- 5}{x} v = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.9

Move the negative in front of the fraction.

$\frac{d v}{d x} - \frac{5}{x} v = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.10

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

$\frac{d v}{d x} - \frac{v \cdot 5}{x} = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.2.1.11

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

$\frac{d v}{d x} - \frac{5 v}{x} = x^{2} {(v^{- \frac{1}{5}})}^{6} (- (5 v^{\frac{6}{5}}))$

Step 7.1.1.3

Simplify the right side.

Tap for more steps...

Step 7.1.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - \frac{5 v}{x} = - 1 \cdot 5 x^{2} {(v^{- \frac{1}{5}})}^{6} v^{\frac{6}{5}}$

Step 7.1.1.3.2

Multiply $- 1$ by $5$ .

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} {(v^{- \frac{1}{5}})}^{6} v^{\frac{6}{5}}$

Step 7.1.1.3.3

Multiply the exponents in ${(v^{- \frac{1}{5}})}^{6}$ .

Tap for more steps...

Step 7.1.1.3.3.1

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

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{- \frac{1}{5} \cdot 6} v^{\frac{6}{5}}$

Step 7.1.1.3.3.2

Multiply $- \frac{1}{5} \cdot 6$ .

Tap for more steps...

Step 7.1.1.3.3.2.1

Multiply $6$ by $- 1$ .

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{- 6 (\frac{1}{5})} v^{\frac{6}{5}}$

Step 7.1.1.3.3.2.2

Combine $- 6$ and $\frac{1}{5}$ .

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{\frac{- 6}{5}} v^{\frac{6}{5}}$

Step 7.1.1.3.3.3

Move the negative in front of the fraction.

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{- \frac{6}{5}} v^{\frac{6}{5}}$

Step 7.1.1.3.4

Multiply $v^{- \frac{6}{5}}$ by $v^{\frac{6}{5}}$ by adding the exponents.

Tap for more steps...

Step 7.1.1.3.4.1

Move $v^{\frac{6}{5}}$ .

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} (v^{\frac{6}{5}} v^{- \frac{6}{5}})$

Step 7.1.1.3.4.2

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

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{\frac{6}{5} - \frac{6}{5}}$

Step 7.1.1.3.4.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{\frac{6 - 6}{5}}$

Step 7.1.1.3.4.4

Subtract $6$ from $6$ .

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{\frac{0}{5}}$

Step 7.1.1.3.4.5

Divide $0$ by $5$ .

$\frac{d v}{d x} - \frac{5 v}{x} = - 5 x^{2} v^{0}$

Step 7.1.1.3.5

Simplify $- 5 x^{2} v^{0}$ .

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

Step 7.1.2

Factor $v$ out of $- \frac{5 v}{x}$ .

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

Step 7.1.3

Reorder $v$ and $- \frac{5}{x}$ .

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

Step 7.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - \frac{5}{x}$ .

Tap for more steps...

Step 7.2.1

Set up the integration.

$e^{\int - \frac{5}{x} d x}$

Step 7.2.2

Integrate $- \frac{5}{x}$ .

Tap for more steps...

Step 7.2.2.1

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

$e^{- \int \frac{5}{x} d x}$

Step 7.2.2.2

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

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

Step 7.2.2.3

Multiply $5$ by $- 1$ .

$e^{- 5 \int \frac{1}{x} d x}$

Step 7.2.2.4

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

$e^{- 5 (\ln (| x |) + C)}$

Step 7.2.2.5

Simplify.

$e^{- 5 \ln (| x |) + C}$

Step 7.2.3

Remove the constant of integration.

$e^{- 5 \ln (x)}$

Step 7.2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 5})}$

Step 7.2.5

Exponentiation and log are inverse functions.

$x^{- 5}$

Step 7.2.6

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

$\frac{1}{x^{5}}$

Step 7.3

Multiply each term by the integrating factor $\frac{1}{x^{5}}$ .

Tap for more steps...

Step 7.3.1

Multiply each term by $\frac{1}{x^{5}}$ .

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

Step 7.3.2

Simplify each term.

Tap for more steps...

Step 7.3.2.1

Combine $\frac{1}{x^{5}}$ and $\frac{d v}{d x}$ .

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

Step 7.3.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.3.2.3

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

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

Step 7.3.2.4

Multiply $- \frac{1}{x^{5}} \cdot \frac{5 v}{x}$ .

Tap for more steps...

Step 7.3.2.4.1

Multiply $\frac{5 v}{x}$ by $\frac{1}{x^{5}}$ .

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

Step 7.3.2.4.2

Multiply $x$ by $x^{5}$ by adding the exponents.

Tap for more steps...

Step 7.3.2.4.2.1

Multiply $x$ by $x^{5}$ .

Tap for more steps...

Step 7.3.2.4.2.1.1

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

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

Step 7.3.2.4.2.1.2

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

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

Step 7.3.2.4.2.2

Add $1$ and $5$ .

$\frac{\frac{d v}{d x}}{x^{5}} - \frac{5 v}{x^{6}} = \frac{1}{x^{5}} (- 5 x^{2})$

Step 7.3.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d v}{d x}}{x^{5}} - \frac{5 v}{x^{6}} = - 5 \frac{1}{x^{5}} x^{2}$

Step 7.3.4

Combine $- 5$ and $\frac{1}{x^{5}}$ .

$\frac{\frac{d v}{d x}}{x^{5}} - \frac{5 v}{x^{6}} = \frac{- 5}{x^{5}} x^{2}$

Step 7.3.5

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

Tap for more steps...

Step 7.3.5.1

Factor $x^{2}$ out of $x^{5}$ .

$\frac{\frac{d v}{d x}}{x^{5}} - \frac{5 v}{x^{6}} = \frac{- 5}{x^{2} x^{3}} x^{2}$

Step 7.3.5.2

Cancel the common factor.

$\frac{\frac{d v}{d x}}{x^{5}} - \frac{5 v}{x^{6}} = \frac{- 5}{x^{2} x^{3}} x^{2}$

Step 7.3.5.3

Rewrite the expression.

$\frac{\frac{d v}{d x}}{x^{5}} - \frac{5 v}{x^{6}} = \frac{- 5}{x^{3}}$

Step 7.3.6

Move the negative in front of the fraction.

$\frac{\frac{d v}{d x}}{x^{5}} - \frac{5 v}{x^{6}} = - \frac{5}{x^{3}}$

Step 7.4

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

$\frac{d}{d x} [\frac{1}{x^{5}} v] = - \frac{5}{x^{3}}$

Step 7.5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{5}} v] d x = \int - \frac{5}{x^{3}} d x$

Step 7.6

Integrate the left side.

$\frac{1}{x^{5}} v = \int - \frac{5}{x^{3}} d x$

Step 7.7

Integrate the right side.

Tap for more steps...

Step 7.7.1

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

$\frac{1}{x^{5}} v = - \int \frac{5}{x^{3}} d x$

Step 7.7.2

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

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

Step 7.7.3

Simplify the expression.

Tap for more steps...

Step 7.7.3.1

Multiply $5$ by $- 1$ .

$\frac{1}{x^{5}} v = - 5 \int \frac{1}{x^{3}} d x$

Step 7.7.3.2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.7.3.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

Tap for more steps...

Step 7.7.3.3.1

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

$\frac{1}{x^{5}} v = - 5 \int x^{3 \cdot - 1} d x$

Step 7.7.3.3.2

Multiply $3$ by $- 1$ .

$\frac{1}{x^{5}} v = - 5 \int x^{- 3} d x$

Step 7.7.4

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

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

Step 7.7.5

Simplify the answer.

Tap for more steps...

Step 7.7.5.1

Simplify.

Tap for more steps...

Step 7.7.5.1.1

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

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

Step 7.7.5.1.2

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

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

Step 7.7.5.2

Simplify.

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

Step 7.7.5.3

Simplify.

Tap for more steps...

Step 7.7.5.3.1

Multiply $- 1$ by $- 5$ .

$\frac{1}{x^{5}} v = 5 \frac{1}{2 x^{2}} + C$

Step 7.7.5.3.2

Combine $5$ and $\frac{1}{2 x^{2}}$ .

$\frac{1}{x^{5}} v = \frac{5}{2 x^{2}} + C$

Step 7.8

Solve for $v$ .

Tap for more steps...

Step 7.8.1

Move all terms containing variables to the left side of the equation.

Tap for more steps...

Step 7.8.1.1

Subtract $\frac{5}{2 x^{2}}$ from both sides of the equation.

$\frac{1}{x^{5}} v - \frac{5}{2 x^{2}} = C$

Step 7.8.1.2

Subtract $C$ from both sides of the equation.

$\frac{1}{x^{5}} v - \frac{5}{2 x^{2}} - C = 0$

Step 7.8.1.3

Combine $\frac{1}{x^{5}}$ and $v$ .

$\frac{v}{x^{5}} - \frac{5}{2 x^{2}} - C = 0$

Step 7.8.2

Move all terms not containing $v$ to the right side of the equation.

Tap for more steps...

Step 7.8.2.1

Add $\frac{5}{2 x^{2}}$ to both sides of the equation.

$\frac{v}{x^{5}} - C = \frac{5}{2 x^{2}}$

Step 7.8.2.2

Add $C$ to both sides of the equation.

$\frac{v}{x^{5}} = \frac{5}{2 x^{2}} + C$

Step 7.8.3

Multiply both sides by $x^{5}$ .

$\frac{v}{x^{5}} x^{5} = (\frac{5}{2 x^{2}} + C) x^{5}$

Step 7.8.4

Simplify.

Tap for more steps...

Step 7.8.4.1

Simplify the left side.

Tap for more steps...

Step 7.8.4.1.1

Cancel the common factor of $x^{5}$ .

Tap for more steps...

Step 7.8.4.1.1.1

Cancel the common factor.

$\frac{v}{x^{5}} x^{5} = (\frac{5}{2 x^{2}} + C) x^{5}$

Step 7.8.4.1.1.2

Rewrite the expression.

$v = (\frac{5}{2 x^{2}} + C) x^{5}$

Step 7.8.4.2

Simplify the right side.

Tap for more steps...

Step 7.8.4.2.1

Simplify $(\frac{5}{2 x^{2}} + C) x^{5}$ .

Tap for more steps...

Step 7.8.4.2.1.1

Apply the distributive property.

$v = \frac{5}{2 x^{2}} x^{5} + C x^{5}$

Step 7.8.4.2.1.2

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

Tap for more steps...

Step 7.8.4.2.1.2.1

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

$v = \frac{5}{x^{2} \cdot 2} x^{5} + C x^{5}$

Step 7.8.4.2.1.2.2

Factor $x^{2}$ out of $x^{5}$ .

$v = \frac{5}{x^{2} \cdot 2} (x^{2} x^{3}) + C x^{5}$

Step 7.8.4.2.1.2.3

Cancel the common factor.

$v = \frac{5}{x^{2} \cdot 2} (x^{2} x^{3}) + C x^{5}$

Step 7.8.4.2.1.2.4

Rewrite the expression.

$v = \frac{5}{2} x^{3} + C x^{5}$

Step 7.8.4.2.1.3

Combine $\frac{5}{2}$ and $x^{3}$ .

$v = \frac{5 x^{3}}{2} + C x^{5}$

Step 7.8.4.2.1.4

Reorder $\frac{5 x^{3}}{2}$ and $C x^{5}$ .

$v = C x^{5} + \frac{5 x^{3}}{2}$

Step 8

Substitute $y^{- 5}$ for $v$ .

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