Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{e^{- y^{2}}}$ .

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

Step 1.2

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

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Step 1.2.1

Factor $e^{- y^{2}}$ out of $x e^{- y^{2}}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Remove unnecessary parentheses.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Combine fractions.

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Step 2.2.1.1

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

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

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Negate the exponent of $e^{- y^{2}}$ and move it out of the denominator.

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

Step 2.2.1.2.2

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

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Step 2.2.1.2.2.1

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

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

Step 2.2.1.2.2.2

Multiply $- y^{2} \cdot - 1$ .

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Step 2.2.1.2.2.2.1

Multiply $- 1$ by $- 1$ .

$\int y e^{1 y^{2}} d y = \int x d x$

Step 2.2.1.2.2.2.2

Multiply $y^{2}$ by $1$ .

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

Step 2.2.2

Let $u = y^{2}$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = y^{2}$ . Find $\frac{d u}{d y}$ .

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Step 2.2.2.1.1

Differentiate $y^{2}$ .

$\frac{d}{d y} [y^{2}]$

Step 2.2.2.1.2

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

$2 y$

Step 2.2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.2.7

Replace all occurrences of $u$ with $y^{2}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

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.

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

Step 3.2.2.1.3.2

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Expand the left side.

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Step 3.4.1

Expand $\ln (e^{y^{2}})$ by moving $y^{2}$ outside the logarithm.

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

Step 3.4.2

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

$y^{2} \cdot 1 = \ln (x^{2} + 2 K)$

Step 3.4.3

Multiply $y^{2}$ by $1$ .

$y^{2} = \ln (x^{2} + 2 K)$

Step 3.5

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

$y = \pm \sqrt{\ln (x^{2} + 2 K)}$

Step 3.6

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

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Step 3.6.1

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

$y = \sqrt{\ln (x^{2} + 2 K)}$

Step 3.6.2

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

$y = - \sqrt{\ln (x^{2} + 2 K)}$

Step 3.6.3

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

$y = \sqrt{\ln (x^{2} + 2 K)}$

$y = - \sqrt{\ln (x^{2} + 2 K)}$

$y = \sqrt{\ln (x^{2} + 2 K)}$

$y = - \sqrt{\ln (x^{2} + 2 K)}$

$y = \sqrt{\ln (x^{2} + 2 K)}$

$y = - \sqrt{\ln (x^{2} + 2 K)}$

Step 4

Simplify the constant of integration.

$y = \sqrt{\ln (x^{2} + K)}$

$y = - \sqrt{\ln (x^{2} + K)}$