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

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $2 y$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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 $y^{2}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

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

Step 3.2.2.1.1.2

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

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Step 3.2.2.1.1.2.1

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

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

Step 3.2.2.1.1.2.2

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

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

Step 3.2.2.1.1.3

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1.1

Cancel the common factor.

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

Step 3.2.2.1.3.1.2

Rewrite the expression.

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

Step 3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.2.1

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

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

Step 3.2.2.1.3.2.2

Cancel the common factor.

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

Step 3.2.2.1.3.2.3

Rewrite the expression.

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

Step 3.3

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

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

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.

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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