Calculus Examples

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

Step 1

Let $v = e^{- y}$ . Substitute $v$ for all occurrences of $e^{- y}$ .

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

Step 2

Find $\frac{d v}{d x}$ by differentiating $e^{- y}$ .

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $- y$ .

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

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 2.1.3

Replace all occurrences of $u$ with $- y$ .

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

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$\frac{d v}{d x} = e^{- y} (- \frac{d}{d x} [y])$

Step 2.3

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

$\frac{d v}{d x} = e^{- y} (- y')$

Step 2.4

Reorder the factors of $e^{- y} (- y')$ .

$\frac{d v}{d x} = - e^{- y} y'$

Step 3

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

$\frac{d v}{d x} = - v y'$

Step 4

Substitute the derivative back in to the differential equation.

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

Step 5

Rewrite the differential equation to fit the Bernoulli technique.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Cancel the common factor of $x$ .

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Step 5.1.2.1.1.1

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

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

Step 5.1.2.1.1.2

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

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

Step 5.1.2.1.1.3

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

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

Step 5.1.2.1.1.4

Cancel the common factor.

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

Step 5.1.2.1.1.5

Rewrite the expression.

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

Step 5.1.2.1.2

Cancel the common factor of $v$ .

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Step 5.1.2.1.2.1

Factor $v$ out of $- v$ .

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

Step 5.1.2.1.2.2

Cancel the common factor.

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

Step 5.1.2.1.2.3

Rewrite the expression.

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

Step 5.1.2.1.3

Multiply $- 1$ by $- 1$ .

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

Step 5.1.2.1.4

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

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

Step 5.1.2.1.5

Multiply $3 (- \frac{v}{x})$ .

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Step 5.1.2.1.5.1

Multiply $- 1$ by $3$ .

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

Step 5.1.2.1.5.2

Combine $- 3$ and $\frac{v}{x}$ .

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

Step 5.1.2.1.6

Move the negative in front of the fraction.

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

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Cancel the common factor of $x$ .

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Step 5.1.3.1.1

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

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

Step 5.1.3.1.2

Factor $x$ out of $4 x v$ .

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

Step 5.1.3.1.3

Cancel the common factor.

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

Step 5.1.3.1.4

Rewrite the expression.

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

Step 5.1.3.2

Multiply $- 1$ by $4$ .

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

Step 5.1.3.3

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

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

Step 5.1.3.4

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

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

Step 5.1.3.5

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

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

Step 5.1.3.6

Add $1$ and $1$ .

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

Step 5.2

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

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

Step 5.3

Reorder $v$ and $- \frac{3}{x}$ .

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

Step 6

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

$v = v^{- 1}$

Step 7

Solve the equation for $v$ .

$y = v^{- 1}$

Step 8

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

$y' = v^{- 1}$

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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 9.4

Differentiate using the Constant Rule.

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Step 9.4.1

Multiply $- 1$ by $1$ .

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

Step 9.4.2

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 9.4.3

Simplify the expression.

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Step 9.4.3.1

Multiply $v$ by $0$ .

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

Step 9.4.3.2

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

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

Step 9.4.3.3

Move the negative in front of the fraction.

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

Step 9.5

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

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

Step 10

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

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

Step 11

Solve the substituted differential equation.

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Step 11.1

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

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Step 11.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v^{2}} - \frac{3}{x} v^{- 1} = - 4 {(v^{- 1})}^{2}$ by $- v^{2}$ to eliminate the fractions.

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Step 11.1.1.1

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

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

Step 11.1.1.2

Simplify the left side.

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Step 11.1.1.2.1

Simplify each term.

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Step 11.1.1.2.1.1

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

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Step 11.1.1.2.1.1.1

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

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

Step 11.1.1.2.1.1.2

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

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

Step 11.1.1.2.1.1.3

Cancel the common factor.

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

Step 11.1.1.2.1.1.4

Rewrite the expression.

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

Step 11.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 11.1.1.2.1.3

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

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

Step 11.1.1.2.1.4

Multiply $v^{- 1}$ by $v^{2}$ by adding the exponents.

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Step 11.1.1.2.1.4.1

Move $v^{2}$ .

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

Step 11.1.1.2.1.4.2

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

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

Step 11.1.1.2.1.4.3

Subtract $1$ from $2$ .

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

Step 11.1.1.2.1.5

Simplify $- \frac{3}{x} v^{1} \cdot - 1$ .

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

Step 11.1.1.2.1.6

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

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

Step 11.1.1.2.1.7

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

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

Step 11.1.1.2.1.8

Multiply $- \frac{3 v}{x} \cdot - 1$ .

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Step 11.1.1.2.1.8.1

Multiply $- 1$ by $- 1$ .

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

Step 11.1.1.2.1.8.2

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

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

Step 11.1.1.3

Simplify the right side.

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Step 11.1.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 11.1.1.3.2

Multiply $- 4$ by $- 1$ .

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

Step 11.1.1.3.3

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

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Step 11.1.1.3.3.1

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

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

Step 11.1.1.3.3.2

Multiply $- 1$ by $2$ .

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

Step 11.1.1.3.4

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

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Step 11.1.1.3.4.1

Move $v^{2}$ .

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

Step 11.1.1.3.4.2

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

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

Step 11.1.1.3.4.3

Subtract $2$ from $2$ .

$\frac{d v}{d x} + \frac{3 v}{x} = 4 v^{0}$

Step 11.1.1.3.5

Simplify $4 v^{0}$ .

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

Step 11.1.2

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

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

Step 11.1.3

Reorder $v$ and $\frac{3}{x}$ .

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

Step 11.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{3}{x}$ .

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Step 11.2.1

Set up the integration.

$e^{\int \frac{3}{x} d x}$

Step 11.2.2

Integrate $\frac{3}{x}$ .

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Step 11.2.2.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{3 \int \frac{1}{x} d x}$

Step 11.2.2.2

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

$e^{3 (\ln (| x |) + C)}$

Step 11.2.2.3

Simplify.

$e^{3 \ln (| x |) + C}$

Step 11.2.3

Remove the constant of integration.

$e^{3 \ln (x)}$

Step 11.2.4

Use the logarithmic power rule.

$e^{\ln (x^{3})}$

Step 11.2.5

Exponentiation and log are inverse functions.

$x^{3}$

Step 11.3

Multiply each term by the integrating factor $x^{3}$ .

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Step 11.3.1

Multiply each term by $x^{3}$ .

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

Step 11.3.2

Simplify each term.

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Step 11.3.2.1

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

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

Step 11.3.2.2

Cancel the common factor of $x$ .

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Step 11.3.2.2.1

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

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

Step 11.3.2.2.2

Cancel the common factor.

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

Step 11.3.2.2.3

Rewrite the expression.

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

Step 11.3.2.3

Rewrite using the commutative property of multiplication.

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

Step 11.3.3

Move $4$ to the left of $x^{3}$ .

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

Step 11.4

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

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

Step 11.5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{3} v] d x = \int 4 x^{3} d x$

Step 11.6

Integrate the left side.

$x^{3} v = \int 4 x^{3} d x$

Step 11.7

Integrate the right side.

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Step 11.7.1

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

$x^{3} v = 4 \int x^{3} d x$

Step 11.7.2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 11.7.3

Simplify the answer.

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Step 11.7.3.1

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

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

Step 11.7.3.2

Simplify.

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Step 11.7.3.2.1

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

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

Step 11.7.3.2.2

Cancel the common factor of $4$ .

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Step 11.7.3.2.2.1

Cancel the common factor.

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

Step 11.7.3.2.2.2

Rewrite the expression.

$x^{3} v = 1 x^{4} + C$

Step 11.7.3.2.3

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

$x^{3} v = x^{4} + C$

Step 11.8

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

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Step 11.8.1

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

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

Step 11.8.2

Simplify the left side.

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Step 11.8.2.1

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

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Step 11.8.2.1.1

Cancel the common factor.

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

Step 11.8.2.1.2

Divide $v$ by $1$ .

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

Step 11.8.3

Simplify the right side.

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Step 11.8.3.1

Cancel the common factor of $x^{4}$ and $x^{3}$ .

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Step 11.8.3.1.1

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

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

Step 11.8.3.1.2

Cancel the common factors.

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Step 11.8.3.1.2.1

Multiply by $1$ .

$v = \frac{x^{3} x}{x^{3} \cdot 1} + \frac{C}{x^{3}}$

Step 11.8.3.1.2.2

Cancel the common factor.

$v = \frac{x^{3} x}{x^{3} \cdot 1} + \frac{C}{x^{3}}$

Step 11.8.3.1.2.3

Rewrite the expression.

$v = \frac{x}{1} + \frac{C}{x^{3}}$

Step 11.8.3.1.2.4

Divide $x$ by $1$ .

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

Step 12

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

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

Step 13

Replace all occurrences of $v$ with $e^{- y}$ .

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

Step 14

Solve for $y$ .

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Step 14.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y}) = \ln (x + \frac{C}{x^{3}})$

Step 14.2

Expand the left side.

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Step 14.2.1

Expand $\ln (e^{y})$ by moving $y$ outside the logarithm.

$y \ln (e) = \ln (x + \frac{C}{x^{3}})$

Step 14.2.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (x + \frac{C}{x^{3}})$

Step 14.2.3

Multiply $y$ by $1$ .

$y = \ln (x + \frac{C}{x^{3}})$