Calculus Examples

$\frac{d y}{d x} + 3 x^{2} y = 4 x^{2} y^{2}$

Step 1

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{2}$ .

$v = y^{- 1}$

Step 2

Solve the equation for $y$ .

$y = v^{- 1}$

Step 3

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

$y' = v^{- 1}$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

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 $g (x) = v$ .

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

Step 4.4

Differentiate using the Constant Rule.

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Step 4.4.1

Multiply $- 1$ by $1$ .

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

Step 4.4.2

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

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

Step 4.4.3

Simplify the expression.

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Step 4.4.3.1

Multiply $v$ by $0$ .

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

Step 4.4.3.2

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

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

Step 4.4.3.3

Move the negative in front of the fraction.

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

Step 4.5

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

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

Step 5

Substitute $- \frac{v'}{v^{2}}$ for $\frac{d y}{d x}$ and $v^{- 1}$ for $y$ in the original equation $\frac{d y}{d x} + 3 x^{2} y = 4 x^{2} y^{2}$ .

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

Step 6

Solve the substituted differential equation.

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Step 6.1

Separate the variables.

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Step 6.1.1

Solve for $\frac{d v}{d x}$ .

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Step 6.1.1.1

Simplify each term.

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Step 6.1.1.1.1

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

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

Step 6.1.1.1.2

Multiply $3 x^{2} \frac{1}{v}$ .

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Step 6.1.1.1.2.1

Combine $3$ and $\frac{1}{v}$ .

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

Step 6.1.1.1.2.2

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

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

Step 6.1.1.1.3

Move $3$ to the left of $x^{2}$ .

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

Step 6.1.1.2

Simplify $4 x^{2} {(v^{- 1})}^{2}$ .

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Step 6.1.1.2.1

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

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Step 6.1.1.2.1.1

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

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

Step 6.1.1.2.1.2

Multiply $- 1$ by $2$ .

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

Step 6.1.1.2.2

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

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

Step 6.1.1.2.3

Multiply $4 x^{2} \frac{1}{v^{2}}$ .

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Step 6.1.1.2.3.1

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

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

Step 6.1.1.2.3.2

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

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

Step 6.1.1.2.4

Move $4$ to the left of $x^{2}$ .

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

Step 6.1.1.3

Subtract $\frac{3 x^{2}}{v}$ from both sides of the equation.

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

Step 6.1.1.4

Divide each term in $- \frac{\frac{d v}{d x}}{v^{2}} = \frac{4 x^{2}}{v^{2}} - \frac{3 x^{2}}{v}$ by $- 1$ and simplify.

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Step 6.1.1.4.1

Divide each term in $- \frac{\frac{d v}{d x}}{v^{2}} = \frac{4 x^{2}}{v^{2}} - \frac{3 x^{2}}{v}$ by $- 1$ .

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

Step 6.1.1.4.2

Simplify the left side.

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Step 6.1.1.4.2.1

Dividing two negative values results in a positive value.

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

Step 6.1.1.4.2.2

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

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

Step 6.1.1.4.3

Simplify the right side.

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Step 6.1.1.4.3.1

Simplify each term.

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Step 6.1.1.4.3.1.1

Move the negative one from the denominator of $\frac{\frac{4 x^{2}}{v^{2}}}{- 1}$ .

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

Step 6.1.1.4.3.1.2

Rewrite $- 1 \cdot \frac{4 x^{2}}{v^{2}}$ as $- \frac{4 x^{2}}{v^{2}}$ .

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

Step 6.1.1.4.3.1.3

Dividing two negative values results in a positive value.

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

Step 6.1.1.4.3.1.4

Divide $\frac{3 x^{2}}{v}$ by $1$ .

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

Step 6.1.1.5

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

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

Step 6.1.1.6

Simplify.

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Step 6.1.1.6.1

Simplify the left side.

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Step 6.1.1.6.1.1

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

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Step 6.1.1.6.1.1.1

Cancel the common factor.

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

Step 6.1.1.6.1.1.2

Rewrite the expression.

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

Step 6.1.1.6.2

Simplify the right side.

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Step 6.1.1.6.2.1

Simplify $(- \frac{4 x^{2}}{v^{2}} + \frac{3 x^{2}}{v}) v^{2}$ .

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Step 6.1.1.6.2.1.1

Apply the distributive property.

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

Step 6.1.1.6.2.1.2

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

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Step 6.1.1.6.2.1.2.1

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

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

Step 6.1.1.6.2.1.2.2

Cancel the common factor.

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

Step 6.1.1.6.2.1.2.3

Rewrite the expression.

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

Step 6.1.1.6.2.1.3

Cancel the common factor of $v$ .

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Step 6.1.1.6.2.1.3.1

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

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

Step 6.1.1.6.2.1.3.2

Cancel the common factor.

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

Step 6.1.1.6.2.1.3.3

Rewrite the expression.

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

Step 6.1.1.6.2.1.4

Simplify the expression.

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Step 6.1.1.6.2.1.4.1

Move $x^{2}$ .

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

Step 6.1.1.6.2.1.4.2

Reorder $- 4 x^{2}$ and $3 v x^{2}$ .

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

Step 6.1.2

Factor $x^{2}$ out of $3 v x^{2} - 4 x^{2}$ .

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Step 6.1.2.1

Factor $x^{2}$ out of $3 v x^{2}$ .

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

Step 6.1.2.2

Factor $x^{2}$ out of $- 4 x^{2}$ .

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

Step 6.1.2.3

Factor $x^{2}$ out of $x^{2} (3 v) + x^{2} \cdot - 4$ .

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

Step 6.1.3

Multiply both sides by $\frac{1}{3 v - 4}$ .

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

Step 6.1.4

Cancel the common factor of $3 v - 4$ .

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Step 6.1.4.1

Factor $3 v - 4$ out of $x^{2} (3 v - 4)$ .

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

Step 6.1.4.2

Cancel the common factor.

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

Step 6.1.4.3

Rewrite the expression.

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

Step 6.1.5

Rewrite the equation.

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

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

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

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Let $u = 3 v - 4$ . Then $d u = 3 d v$ , so $\frac{1}{3} d u = d v$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.2.1.1

Let $u = 3 v - 4$ . Find $\frac{d u}{d v}$ .

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Step 6.2.2.1.1.1

Differentiate $3 v - 4$ .

$\frac{d}{d v} [3 v - 4]$

Step 6.2.2.1.1.2

By the Sum Rule, the derivative of $3 v - 4$ with respect to $v$ is $\frac{d}{d v} [3 v] + \frac{d}{d v} [- 4]$ .

$\frac{d}{d v} [3 v] + \frac{d}{d v} [- 4]$

Step 6.2.2.1.1.3

Evaluate $\frac{d}{d v} [3 v]$ .

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Step 6.2.2.1.1.3.1

Since $3$ is constant with respect to $v$ , the derivative of $3 v$ with respect to $v$ is $3 \frac{d}{d v} [v]$ .

$3 \frac{d}{d v} [v] + \frac{d}{d v} [- 4]$

Step 6.2.2.1.1.3.2

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

$3 \cdot 1 + \frac{d}{d v} [- 4]$

Step 6.2.2.1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d v} [- 4]$

Step 6.2.2.1.1.4

Differentiate using the Constant Rule.

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Step 6.2.2.1.1.4.1

Since $- 4$ is constant with respect to $v$ , the derivative of $- 4$ with respect to $v$ is $0$ .

$3 + 0$

Step 6.2.2.1.1.4.2

Add $3$ and $0$ .

$3$

Step 6.2.2.1.2

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

$\int \frac{1}{u} \cdot \frac{1}{3} d u = \int x^{2} d x$

Step 6.2.2.2

Simplify.

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Step 6.2.2.2.1

Multiply $\frac{1}{u}$ by $\frac{1}{3}$ .

$\int \frac{1}{u \cdot 3} d u = \int x^{2} d x$

Step 6.2.2.2.2

Move $3$ to the left of $u$ .

$\int \frac{1}{3 u} d u = \int x^{2} d x$

Step 6.2.2.3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{1}{u} d u = \int x^{2} d x$

Step 6.2.2.4

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

$\frac{1}{3} (\ln (| u |) + C_{1}) = \int x^{2} d x$

Step 6.2.2.5

Simplify.

$\frac{1}{3} \ln (| u |) + C_{1} = \int x^{2} d x$

Step 6.2.2.6

Replace all occurrences of $u$ with $3 v - 4$ .

$\frac{1}{3} \ln (| 3 v - 4 |) + C_{1} = \int x^{2} d x$

Step 6.2.3

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

$\frac{1}{3} \ln (| 3 v - 4 |) + C_{1} = \frac{1}{3} x^{3} + C_{2}$

Step 6.2.4

Group the constant of integration on the right side as $C$ .

$\frac{1}{3} \ln (| 3 v - 4 |) = \frac{1}{3} x^{3} + C$

Step 6.3

Solve for $v$ .

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Step 6.3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \ln (| 3 v - 4 |)) = 3 (\frac{1}{3} x^{3} + C)$

Step 6.3.2

Simplify both sides of the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Simplify $3 (\frac{1}{3} \ln (| 3 v - 4 |))$ .

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Step 6.3.2.1.1.1

Combine $\frac{1}{3}$ and $\ln (| 3 v - 4 |)$ .

$3 \frac{\ln (| 3 v - 4 |)}{3} = 3 (\frac{1}{3} x^{3} + C)$

Step 6.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.1.1.2.1

Cancel the common factor.

$3 \frac{\ln (| 3 v - 4 |)}{3} = 3 (\frac{1}{3} x^{3} + C)$

Step 6.3.2.1.1.2.2

Rewrite the expression.

$\ln (| 3 v - 4 |) = 3 (\frac{1}{3} x^{3} + C)$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Simplify $3 (\frac{1}{3} x^{3} + C)$ .

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Step 6.3.2.2.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$\ln (| 3 v - 4 |) = 3 (\frac{x^{3}}{3} + C)$

Step 6.3.2.2.1.2

Apply the distributive property.

$\ln (| 3 v - 4 |) = 3 \frac{x^{3}}{3} + 3 C$

Step 6.3.2.2.1.3

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.3.1

Cancel the common factor.

$\ln (| 3 v - 4 |) = 3 \frac{x^{3}}{3} + 3 C$

Step 6.3.2.2.1.3.2

Rewrite the expression.

$\ln (| 3 v - 4 |) = x^{3} + 3 C$

Step 6.3.3

To solve for $v$ , rewrite the equation using properties of logarithms.

$e^{\ln (| 3 v - 4 |)} = e^{x^{3} + 3 C}$

Step 6.3.4

Rewrite $\ln (| 3 v - 4 |) = x^{3} + 3 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{x^{3} + 3 C} = | 3 v - 4 |$

Step 6.3.5

Solve for $v$ .

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Step 6.3.5.1

Rewrite the equation as $| 3 v - 4 | = e^{x^{3} + 3 C}$ .

$| 3 v - 4 | = e^{x^{3} + 3 C}$

Step 6.3.5.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$3 v - 4 = \pm e^{x^{3} + 3 C}$

Step 6.3.5.3

Add $4$ to both sides of the equation.

$3 v = \pm e^{x^{3} + 3 C} + 4$

Step 6.3.5.4

Divide each term in $3 v = \pm e^{x^{3} + 3 C} + 4$ by $3$ and simplify.

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Step 6.3.5.4.1

Divide each term in $3 v = \pm e^{x^{3} + 3 C} + 4$ by $3$ .

$\frac{3 v}{3} = \frac{\pm e^{x^{3} + 3 C}}{3} + \frac{4}{3}$

Step 6.3.5.4.2

Simplify the left side.

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Step 6.3.5.4.2.1

Cancel the common factor of $3$ .

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Step 6.3.5.4.2.1.1

Cancel the common factor.

$\frac{3 v}{3} = \frac{\pm e^{x^{3} + 3 C}}{3} + \frac{4}{3}$

Step 6.3.5.4.2.1.2

Divide $v$ by $1$ .

$v = \frac{\pm e^{x^{3} + 3 C}}{3} + \frac{4}{3}$

Step 6.3.5.4.3

Simplify the right side.

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Step 6.3.5.4.3.1

Combine the numerators over the common denominator.

$v = \frac{\pm e^{x^{3} + 3 C} + 4}{3}$

Step 6.4

Group the constant terms together.

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Step 6.4.1

Simplify the constant of integration.

$v = \frac{\pm e^{x^{3} + C} + 4}{3}$

Step 6.4.2

Rewrite $e^{x^{3} + C}$ as $e^{x^{3}} e^{C}$ .

$v = \frac{\pm e^{x^{3}} e^{C} + 4}{3}$

Step 6.4.3

Reorder $e^{x^{3}}$ and $e^{C}$ .

$v = \frac{\pm e^{C} e^{x^{3}} + 4}{3}$

Step 6.4.4

Combine constants with the plus or minus.

$v = \frac{C e^{x^{3}} + 4}{3}$

Step 7

Substitute $y^{- 1}$ for $v$ .

$y^{- 1} = \frac{C e^{x^{3}} + 4}{3}$