Calculus Examples

$x \frac{d y}{d x} + 6 y = 3 x y^{\frac{4}{3}}$

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Divide each term in $x \frac{d y}{d x} + 6 y = 3 x y^{\frac{4}{3}}$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} + \frac{6 y}{x} = \frac{3 x y^{\frac{4}{3}}}{x}$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} + \frac{6 y}{x} = \frac{3 x y^{\frac{4}{3}}}{x}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{6 y}{x} = \frac{3 x y^{\frac{4}{3}}}{x}$

Step 1.3

Cancel the common factor.

$\frac{d y}{d x} + \frac{6 y}{x} = \frac{3 x y^{\frac{4}{3}}}{x}$

Step 1.4

Divide $3 y^{\frac{4}{3}}$ by $1$ .

$\frac{d y}{d x} + \frac{6 y}{x} = 3 y^{\frac{4}{3}}$

Step 1.5

Factor $y$ out of $\frac{6 y}{x}$ .

$\frac{d y}{d x} + y \frac{6}{x} = 3 y^{\frac{4}{3}}$

Step 1.6

Reorder $y$ and $\frac{6}{x}$ .

$\frac{d y}{d x} + \frac{6}{x} y = 3 y^{\frac{4}{3}}$

Step 2

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{\frac{4}{3}}$ .

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

Step 3

Solve the equation for $y$ .

$y = v^{- 3}$

Step 4

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

$y' = v^{- 3}$

Step 5

Take the derivative of $v^{- 3}$ with respect to $x$ .

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Step 5.1

Take the derivative of $v^{- 3}$ .

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

Step 5.2

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

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

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 $g (x) = v^{3}$ .

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

Step 5.4

Differentiate using the Constant Rule.

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Step 5.4.1

Multiply $- 1$ by $1$ .

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

Step 5.4.2

Multiply the exponents in ${(v^{3})}^{2}$ .

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Step 5.4.2.1

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

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

Step 5.4.2.2

Multiply $3$ by $2$ .

$y' = \frac{v^{3} \frac{d}{d x} [1] - \frac{d}{d x} [v^{3}]}{v^{6}}$

Step 5.4.3

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

$y' = \frac{v^{3} \cdot 0 - \frac{d}{d x} [v^{3}]}{v^{6}}$

Step 5.4.4

Simplify the expression.

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Step 5.4.4.1

Multiply $v^{3}$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v^{3}]}{v^{6}}$

Step 5.4.4.2

Subtract $\frac{d}{d x} [v^{3}]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v^{3}]}{v^{6}}$

Step 5.4.4.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v^{3}]}{v^{6}}$

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 $f (x) = x^{3}$ and $g (x) = v$ .

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Step 5.5.1

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

$y' = - \frac{\frac{d}{d u} [u^{3}] \frac{d}{d x} [v]}{v^{6}}$

Step 5.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$y' = - \frac{3 u^{2} \frac{d}{d x} [v]}{v^{6}}$

Step 5.5.3

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

$y' = - \frac{3 v^{2} \frac{d}{d x} [v]}{v^{6}}$

Step 5.6

Cancel the common factor of $v^{2}$ and $v^{6}$ .

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Step 5.6.1

Factor $v^{2}$ out of $3 v^{2} \frac{d}{d x} [v]$ .

$y' = - \frac{v^{2} (3 \frac{d}{d x} [v])}{v^{6}}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $v^{2}$ out of $v^{6}$ .

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

Step 5.6.2.2

Cancel the common factor.

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

Step 5.6.2.3

Rewrite the expression.

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

Step 5.7

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

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

Step 6

Substitute $- \frac{3 v'}{v^{4}}$ for $\frac{d y}{d x}$ and $v^{- 3}$ for $y$ in the original equation $\frac{d y}{d x} + \frac{6}{x} y = 3 y^{\frac{4}{3}}$ .

$- \frac{3 v'}{v^{4}} + \frac{6}{x} v^{- 3} = 3 {(v^{- 3})}^{\frac{4}{3}}$

Step 7

Solve the substituted differential equation.

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Step 7.1

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

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Step 7.1.1

Multiply each term in $- \frac{3 \frac{d v}{d x}}{v^{4}} + \frac{6}{x} v^{- 3} = 3 {(v^{- 3})}^{\frac{4}{3}}$ by $- \frac{v^{4}}{3}$ to eliminate the fractions.

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Step 7.1.1.1

Multiply each term in $- \frac{3 \frac{d v}{d x}}{v^{4}} + \frac{6}{x} v^{- 3} = 3 {(v^{- 3})}^{\frac{4}{3}}$ by $- \frac{v^{4}}{3}$ .

$- \frac{3 \frac{d v}{d x}}{v^{4}} (- \frac{v^{4}}{3}) + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2

Simplify the left side.

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Step 7.1.1.2.1

Simplify each term.

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Step 7.1.1.2.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.2.1.1.1

Move the leading negative in $- \frac{3 \frac{d v}{d x}}{v^{4}}$ into the numerator.

$\frac{- 3 \frac{d v}{d x}}{v^{4}} (- \frac{v^{4}}{3}) + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.1.2

Move the leading negative in $- \frac{v^{4}}{3}$ into the numerator.

$\frac{- 3 \frac{d v}{d x}}{v^{4}} \cdot \frac{- v^{4}}{3} + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.1.3

Factor $3$ out of $- 3 \frac{d v}{d x}$ .

$\frac{3 (- \frac{d v}{d x})}{v^{4}} \cdot \frac{- v^{4}}{3} + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.1.4

Cancel the common factor.

$\frac{3 (- \frac{d v}{d x})}{v^{4}} \cdot \frac{- v^{4}}{3} + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.1.5

Rewrite the expression.

$\frac{- \frac{d v}{d x}}{v^{4}} (- v^{4}) + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.2

Cancel the common factor of $v^{4}$ .

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Step 7.1.1.2.1.2.1

Factor $v^{4}$ out of $- v^{4}$ .

$\frac{- \frac{d v}{d x}}{v^{4}} (v^{4} \cdot - 1) + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.2.2

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{v^{4}} (v^{4} \cdot - 1) + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.2.3

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.3

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.4

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

$\frac{d v}{d x} + \frac{6}{x} v^{- 3} (- \frac{v^{4}}{3}) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.5

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - \frac{6}{x} v^{- 3} \frac{v^{4}}{3} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.6

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

$\frac{d v}{d x} - \frac{6}{x} \cdot \frac{1}{v^{3}} \frac{v^{4}}{3} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.7

Multiply $\frac{1}{v^{3}}$ by $\frac{6}{x}$ .

$\frac{d v}{d x} - \frac{6}{v^{3} x} \cdot \frac{v^{4}}{3} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.8

Cancel the common factor of $3$ .

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Step 7.1.1.2.1.8.1

Move the leading negative in $- \frac{6}{v^{3} x}$ into the numerator.

$\frac{d v}{d x} + \frac{- 6}{v^{3} x} \cdot \frac{v^{4}}{3} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.8.2

Factor $3$ out of $- 6$ .

$\frac{d v}{d x} + \frac{3 (- 2)}{v^{3} x} \cdot \frac{v^{4}}{3} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.8.3

Cancel the common factor.

$\frac{d v}{d x} + \frac{3 \cdot - 2}{v^{3} x} \cdot \frac{v^{4}}{3} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.8.4

Rewrite the expression.

$\frac{d v}{d x} + \frac{- 2}{v^{3} x} v^{4} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.9

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

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Step 7.1.1.2.1.9.1

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

$\frac{d v}{d x} + \frac{- 2}{v^{3} (x)} v^{4} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.9.2

Factor $v^{3}$ out of $v^{4}$ .

$\frac{d v}{d x} + \frac{- 2}{v^{3} (x)} (v^{3} v) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.9.3

Cancel the common factor.

$\frac{d v}{d x} + \frac{- 2}{v^{3} x} (v^{3} v) = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.9.4

Rewrite the expression.

$\frac{d v}{d x} + \frac{- 2}{x} v = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.10

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

$\frac{d v}{d x} + \frac{- 2 v}{x} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.2.1.11

Move the negative in front of the fraction.

$\frac{d v}{d x} - \frac{2 v}{x} = 3 {(v^{- 3})}^{\frac{4}{3}} (- \frac{v^{4}}{3})$

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

Cancel the common factor of $3$ .

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Step 7.1.1.3.1.1

Move the leading negative in $- \frac{v^{4}}{3}$ into the numerator.

$\frac{d v}{d x} - \frac{2 v}{x} = 3 {(v^{- 3})}^{\frac{4}{3}} \frac{- v^{4}}{3}$

Step 7.1.1.3.1.2

Factor $3$ out of $3 {(v^{- 3})}^{\frac{4}{3}}$ .

$\frac{d v}{d x} - \frac{2 v}{x} = 3 ({(v^{- 3})}^{\frac{4}{3}}) \frac{- v^{4}}{3}$

Step 7.1.1.3.1.3

Cancel the common factor.

$\frac{d v}{d x} - \frac{2 v}{x} = 3 {(v^{- 3})}^{\frac{4}{3}} \frac{- v^{4}}{3}$

Step 7.1.1.3.1.4

Rewrite the expression.

$\frac{d v}{d x} - \frac{2 v}{x} = {(v^{- 3})}^{\frac{4}{3}} (- v^{4})$

Step 7.1.1.3.2

Simplify the expression.

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Step 7.1.1.3.2.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - \frac{2 v}{x} = - {(v^{- 3})}^{\frac{4}{3}} v^{4}$

Step 7.1.1.3.2.2

Multiply the exponents in ${(v^{- 3})}^{\frac{4}{3}}$ .

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Step 7.1.1.3.2.2.1

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

$\frac{d v}{d x} - \frac{2 v}{x} = - v^{- 3 (\frac{4}{3})} v^{4}$

Step 7.1.1.3.2.2.2

Cancel the common factor of $3$ .

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Step 7.1.1.3.2.2.2.1

Factor $3$ out of $- 3$ .

$\frac{d v}{d x} - \frac{2 v}{x} = - v^{3 (- 1) \frac{4}{3}} v^{4}$

Step 7.1.1.3.2.2.2.2

Cancel the common factor.

$\frac{d v}{d x} - \frac{2 v}{x} = - v^{3 \cdot - 1 \frac{4}{3}} v^{4}$

Step 7.1.1.3.2.2.2.3

Rewrite the expression.

$\frac{d v}{d x} - \frac{2 v}{x} = - v^{- 1 \cdot 4} v^{4}$

Step 7.1.1.3.2.2.3

Multiply $- 1$ by $4$ .

$\frac{d v}{d x} - \frac{2 v}{x} = - v^{- 4} v^{4}$

Step 7.1.1.3.3

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

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Step 7.1.1.3.3.1

Move $v^{4}$ .

$\frac{d v}{d x} - \frac{2 v}{x} = - (v^{4} v^{- 4})$

Step 7.1.1.3.3.2

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

$\frac{d v}{d x} - \frac{2 v}{x} = - v^{4 - 4}$

Step 7.1.1.3.3.3

Subtract $4$ from $4$ .

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

Step 7.1.1.3.4

Simplify $- v^{0}$ .

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

Step 7.1.2

Factor $v$ out of $- \frac{2 v}{x}$ .

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

Step 7.1.3

Reorder $v$ and $- \frac{2}{x}$ .

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

Step 7.2

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

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Step 7.2.1

Set up the integration.

$e^{\int - \frac{2}{x} d x}$

Step 7.2.2

Integrate $- \frac{2}{x}$ .

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Step 7.2.2.1

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

$e^{- \int \frac{2}{x} d x}$

Step 7.2.2.2

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

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

Step 7.2.2.3

Multiply $2$ by $- 1$ .

$e^{- 2 \int \frac{1}{x} d x}$

Step 7.2.2.4

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

$e^{- 2 (\ln (| x |) + C)}$

Step 7.2.2.5

Simplify.

$e^{- 2 \ln (| x |) + C}$

Step 7.2.3

Remove the constant of integration.

$e^{- 2 \ln (x)}$

Step 7.2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 2})}$

Step 7.2.5

Exponentiation and log are inverse functions.

$x^{- 2}$

Step 7.2.6

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

$\frac{1}{x^{2}}$

Step 7.3

Multiply each term by the integrating factor $\frac{1}{x^{2}}$ .

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Step 7.3.1

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

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.3.2.3

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

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

Step 7.3.2.4

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

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Step 7.3.2.4.1

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

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

Step 7.3.2.4.2

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

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Step 7.3.2.4.2.1

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

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Step 7.3.2.4.2.1.1

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

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

Step 7.3.2.4.2.1.2

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

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

Step 7.3.2.4.2.2

Add $1$ and $2$ .

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

Step 7.3.3

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

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

Step 7.3.4

Move the negative in front of the fraction.

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

Step 7.4

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

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

Step 7.5

Set up an integral on each side.

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

Step 7.6

Integrate the left side.

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

Step 7.7

Integrate the right side.

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Step 7.7.1

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

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

Step 7.7.2

Apply basic rules of exponents.

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Step 7.7.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.7.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.7.2.2.1

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

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

Step 7.7.2.2.2

Multiply $2$ by $- 1$ .

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

Step 7.7.3

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

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

Step 7.7.4

Simplify the answer.

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Step 7.7.4.1

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

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

Step 7.7.4.2

Simplify.

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Step 7.7.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 7.7.4.2.2

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

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

Step 7.8

Solve for $v$ .

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Step 7.8.1

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

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Step 7.8.1.1

Subtract $\frac{1}{x}$ from both sides of the equation.

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

Step 7.8.1.2

Subtract $C$ from both sides of the equation.

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

Step 7.8.1.3

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

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

Step 7.8.2

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

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Step 7.8.2.1

Add $\frac{1}{x}$ to both sides of the equation.

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

Step 7.8.2.2

Add $C$ to both sides of the equation.

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

Step 7.8.3

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

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

Step 7.8.4

Simplify.

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Step 7.8.4.1

Simplify the left side.

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Step 7.8.4.1.1

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

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Step 7.8.4.1.1.1

Cancel the common factor.

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

Step 7.8.4.1.1.2

Rewrite the expression.

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

Step 7.8.4.2

Simplify the right side.

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Step 7.8.4.2.1

Simplify $(\frac{1}{x} + C) x^{2}$ .

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Step 7.8.4.2.1.1

Apply the distributive property.

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

Step 7.8.4.2.1.2

Cancel the common factor of $x$ .

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Step 7.8.4.2.1.2.1

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

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

Step 7.8.4.2.1.2.2

Cancel the common factor.

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

Step 7.8.4.2.1.2.3

Rewrite the expression.

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

Step 7.8.4.2.1.3

Reorder $x$ and $C x^{2}$ .

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

Step 8

Substitute $y^{- \frac{1}{3}}$ for $v$ .

$y^{- \frac{1}{3}} = C x^{2} + x$