Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

Cancel the common factor of $y$ .

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Step 1.2.2.1

Factor $y$ out of $- y$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

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

Step 1.3

Rewrite the equation.

$y d y = - x e^{- x^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int - x e^{- x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{2} y^{2} + C_{1} = - \int x e^{- x^{2}} d x$

Step 2.3.2

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

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

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 $g (x) = - x^{2}$ .

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Step 2.3.2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- x^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x^{2}]$

Step 2.3.2.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- x^{2}]$

Step 2.3.2.1.2.3

Replace all occurrences of $u_{1}$ with $- x^{2}$ .

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

Step 2.3.2.1.3

Differentiate.

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Step 2.3.2.1.3.1

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

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

Step 2.3.2.1.3.2

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

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

Step 2.3.2.1.3.3

Multiply $2$ by $- 1$ .

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

Step 2.3.2.1.4

Simplify.

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Step 2.3.2.1.4.1

Reorder the factors of $e^{- x^{2}} (- 2 x)$ .

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

Step 2.3.2.1.4.2

Reorder factors in $- 2 e^{- x^{2}} x$ .

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

Step 2.3.2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - \int \frac{1}{- 2} d u_{2}$

Step 2.3.3

Move the negative in front of the fraction.

$\frac{1}{2} y^{2} + C_{1} = - \int - \frac{1}{2} d u_{2}$

Step 2.3.4

Apply the constant rule.

$\frac{1}{2} y^{2} + C_{1} = - (- \frac{1}{2} u_{2} + C_{2})$

Step 2.3.5

Simplify the answer.

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Step 2.3.5.1

Simplify.

$\frac{1}{2} y^{2} + C_{1} = - (- \frac{1}{2} u_{2}) + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

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

$\frac{1}{2} y^{2} + C_{1} = - - \frac{u_{2}}{2} + C_{2}$

Step 2.3.5.2.2

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} y^{2} + C_{1} = 1 \frac{u_{2}}{2} + C_{2}$

Step 2.3.5.2.3

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

$\frac{1}{2} y^{2} + C_{1} = \frac{u_{2}}{2} + C_{2}$

Step 2.3.5.3

Replace all occurrences of $u_{2}$ with $e^{- x^{2}}$ .

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

Step 2.3.5.4

Reorder terms.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{2} e^{- x^{2}} + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\frac{1}{2} e^{- x^{2}} + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{2} e^{- x^{2}} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{2} e^{- x^{2}} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{2} e^{- x^{2}} + K)$ .

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Step 3.2.2.1.1

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

$y^{2} = 2 (\frac{e^{- x^{2}}}{2} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{e^{- x^{2}}}{2} + 2 K$

Step 3.2.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1

Cancel the common factor.

$y^{2} = 2 \frac{e^{- x^{2}}}{2} + 2 K$

Step 3.2.2.1.3.2

Rewrite the expression.

$y^{2} = e^{- x^{2}} + 2 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{e^{- x^{2}} + 2 K}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{e^{- x^{2}} + 2 K}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{e^{- x^{2}} + 2 K}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{e^{- x^{2}} + 2 K}$

$y = - \sqrt{e^{- x^{2}} + 2 K}$

$y = \sqrt{e^{- x^{2}} + 2 K}$

$y = - \sqrt{e^{- x^{2}} + 2 K}$

$y = \sqrt{e^{- x^{2}} + 2 K}$

$y = - \sqrt{e^{- x^{2}} + 2 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt{e^{- x^{2}} + K}$

$y = - \sqrt{e^{- x^{2}} + K}$