Calculus Examples

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

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

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

Step 6

Solve the substituted differential equation.

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Step 6.1

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

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Step 6.1.1

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

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Step 6.1.1.1

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

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Step 6.1.1.1.1

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

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

Step 6.1.1.1.2

Simplify the left side.

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Step 6.1.1.1.2.1

Simplify each term.

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Step 6.1.1.1.2.1.1

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

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Step 6.1.1.1.2.1.1.1

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

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

Step 6.1.1.1.2.1.1.2

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

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

Step 6.1.1.1.2.1.1.3

Cancel the common factor.

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

Step 6.1.1.1.2.1.1.4

Rewrite the expression.

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

Step 6.1.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 6.1.1.1.2.1.3

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

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

Step 6.1.1.1.2.1.4

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

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Step 6.1.1.1.2.1.4.1

Move $v^{2}$ .

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

Step 6.1.1.1.2.1.4.2

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

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

Step 6.1.1.1.2.1.4.3

Subtract $1$ from $2$ .

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

Step 6.1.1.1.2.1.5

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

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

Step 6.1.1.1.2.1.6

Multiply $- 1$ by $2$ .

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

Step 6.1.1.1.3

Simplify the right side.

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Step 6.1.1.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.1.1.3.2

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

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Step 6.1.1.1.3.2.1

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

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

Step 6.1.1.1.3.2.2

Multiply $- 1$ by $2$ .

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

Step 6.1.1.1.3.3

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

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Step 6.1.1.1.3.3.1

Move $v^{2}$ .

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

Step 6.1.1.1.3.3.2

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

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

Step 6.1.1.1.3.3.3

Subtract $2$ from $2$ .

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

Step 6.1.1.1.3.4

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

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

Step 6.1.1.2

Reorder terms.

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

Step 6.1.2

Factor $v$ out of $- 2 v x$ .

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

Step 6.1.3

Reorder $v$ and $- 2 x$ .

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

Step 6.2

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

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Step 6.2.1

Set up the integration.

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

Step 6.2.2

Integrate $- 2 x$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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

Step 6.2.2.3

Simplify the answer.

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Step 6.2.2.3.1

Rewrite $- 2 (\frac{1}{2} x^{2} + C)$ as $- 2 (\frac{1}{2}) x^{2} + C$ .

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

Step 6.2.2.3.2

Simplify.

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Step 6.2.2.3.2.1

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

$e^{\frac{- 2}{2} x^{2} + C}$

Step 6.2.2.3.2.2

Cancel the common factor of $- 2$ and $2$ .

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Step 6.2.2.3.2.2.1

Factor $2$ out of $- 2$ .

$e^{\frac{2 \cdot - 1}{2} x^{2} + C}$

Step 6.2.2.3.2.2.2

Cancel the common factors.

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Step 6.2.2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 6.2.2.3.2.2.2.2

Cancel the common factor.

$e^{\frac{2 \cdot - 1}{2 \cdot 1} x^{2} + C}$

Step 6.2.2.3.2.2.2.3

Rewrite the expression.

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

Step 6.2.2.3.2.2.2.4

Divide $- 1$ by $1$ .

$e^{- x^{2} + C}$

Step 6.2.3

Remove the constant of integration.

$e^{- x^{2}}$

Step 6.3

Multiply each term by the integrating factor $e^{- x^{2}}$ .

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Step 6.3.1

Multiply each term by $e^{- x^{2}}$ .

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

Step 6.3.2

Rewrite using the commutative property of multiplication.

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

Step 6.3.3

Rewrite using the commutative property of multiplication.

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

Step 6.3.4

Reorder factors in $e^{- x^{2}} \frac{d v}{d x} - 2 e^{- x^{2}} (x v) = - e^{- x^{2}} x^{2}$ .

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

Step 6.4

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

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

Step 6.5

Set up an integral on each side.

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

Step 6.6

Integrate the left side.

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

Step 6.7

Integrate the right side.

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Step 6.7.1

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

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

Step 6.7.2

Let $u_{1} = - x^{2}$ . Then $d u_{1} = - 2 x d x$ , so $- \frac{1}{2} d u_{1} = x d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 6.7.2.1

Let $u_{1} = - x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 6.7.2.1.1

Differentiate $- x^{2}$ .

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

Step 6.7.2.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 6.7.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- (2 x)$

Step 6.7.2.1.4

Multiply $2$ by $- 1$ .

$- 2 x$

Step 6.7.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 6.7.3

Simplify.

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Step 6.7.3.1

Move the negative in front of the fraction.

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

Step 6.7.3.2

Combine $\sqrt{- u_{1}}$ and $\frac{1}{2}$ .

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

Step 6.7.3.3

Combine $e^{u_{1}}$ and $\frac{\sqrt{- u_{1}}}{2}$ .

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

Step 6.7.4

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

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

Step 6.7.5

Simplify.

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Step 6.7.5.1

Multiply $- 1$ by $- 1$ .

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

Step 6.7.5.2

Multiply $\int \frac{e^{u_{1}} \sqrt{- u_{1}}}{2} d u_{1}$ by $1$ .

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

Step 6.7.6

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

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

Step 6.7.7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = e^{u_{1}}$ and $d v = \sqrt{- u_{1}}$ .

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

Step 6.7.8

Simplify.

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Step 6.7.8.1

Combine ${u_{2}}^{\frac{3}{2}}$ and $\frac{2}{3}$ .

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

Step 6.7.8.2

Combine $e^{u_{1}}$ and $\frac{{u_{2}}^{\frac{3}{2}} \cdot 2}{3}$ .

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

Step 6.7.8.3

Move $2$ to the left of ${u_{2}}^{\frac{3}{2}}$ .

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

Step 6.7.8.4

Move $2$ to the left of $e^{u_{1}}$ .

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

Step 6.7.8.5

Combine ${u_{2}}^{\frac{3}{2}}$ and $\frac{2}{3}$ .

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

Step 6.7.8.6

Combine $e^{u_{1}}$ and $\frac{{u_{2}}^{\frac{3}{2}} \cdot 2}{3}$ .

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

Step 6.7.8.7

Move $2$ to the left of ${u_{2}}^{\frac{3}{2}}$ .

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

Step 6.7.8.8

Move $2$ to the left of $e^{u_{1}}$ .

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

Step 6.7.9

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

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

Step 6.7.10

Simplify.

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Step 6.7.10.1

Multiply $- 1$ by $- 1$ .

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

Step 6.7.10.2

Multiply $\int \frac{2 e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} d u_{1}$ by $1$ .

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

Step 6.7.11

Since $\frac{2 {u_{2}}^{\frac{3}{2}}}{3}$ is constant with respect to $u_{1}$ , move $\frac{2 {u_{2}}^{\frac{3}{2}}}{3}$ out of the integral.

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

Step 6.7.12

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{2 e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} + \frac{2 {u_{2}}^{\frac{3}{2}}}{3} (e^{u_{1}} + C))$

Step 6.7.13

Simplify.

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Step 6.7.13.1

Rewrite $\frac{1}{2} (- \frac{2 e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} + \frac{2 {u_{2}}^{\frac{3}{2}}}{3} (e^{u_{1}} + C))$ as $\frac{1}{2} (- \frac{2}{3} e^{u_{1}} {u_{2}}^{\frac{3}{2}} + \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}}) + C$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{2}{3} e^{u_{1}} {u_{2}}^{\frac{3}{2}} + \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}}) + C$

Step 6.7.13.2

Simplify.

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Step 6.7.13.2.1

Combine $e^{u_{1}}$ and $\frac{2}{3}$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{e^{u_{1}} \cdot 2}{3} {u_{2}}^{\frac{3}{2}} + \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}}) + C$

Step 6.7.13.2.2

Combine ${u_{2}}^{\frac{3}{2}}$ and $\frac{e^{u_{1}} \cdot 2}{3}$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{{u_{2}}^{\frac{3}{2}} (e^{u_{1}} \cdot 2)}{3} + \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}}) + C$

Step 6.7.13.2.3

Move $2$ to the left of $e^{u_{1}}$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{{u_{2}}^{\frac{3}{2}} (2 \cdot e^{u_{1}})}{3} + \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}}) + C$

Step 6.7.13.2.4

Move $2$ to the left of ${u_{2}}^{\frac{3}{2}}$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{2 \cdot {u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3} + \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}}) + C$

Step 6.7.13.2.5

Combine $\frac{2}{3}$ and ${u_{2}}^{\frac{3}{2}}$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{2 {u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3} + \frac{2 {u_{2}}^{\frac{3}{2}}}{3} e^{u_{1}}) + C$

Step 6.7.13.2.6

Combine $\frac{2 {u_{2}}^{\frac{3}{2}}}{3}$ and $e^{u_{1}}$ .

$e^{- x^{2}} v = \frac{1}{2} (- \frac{2 {u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3} + \frac{2 {u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3}) + C$

Step 6.7.13.2.7

Add $- \frac{2 {u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3}$ and $\frac{2 {u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3}$ .

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

Step 6.7.13.2.8

Multiply $\frac{1}{2}$ by $0$ .

$e^{- x^{2}} v = 0 + C$

Step 6.7.13.2.9

Add $0$ and $C$ .

$e^{- x^{2}} v = C$

Step 6.8

Divide each term in $e^{- x^{2}} v = C$ by $e^{- x^{2}}$ and simplify.

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Step 6.8.1

Divide each term in $e^{- x^{2}} v = C$ by $e^{- x^{2}}$ .

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

Step 6.8.2

Simplify the left side.

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Step 6.8.2.1

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

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Step 6.8.2.1.1

Cancel the common factor.

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

Step 6.8.2.1.2

Divide $v$ by $1$ .

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

Step 7

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

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