Calculus Examples

$\frac{d y}{d x} + x y = x^{2}$ , $y (0) = 0$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int x d x}$

Step 1.2

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

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

Step 1.3

Remove the constant of integration.

$e^{\frac{1}{2} x^{2}}$

Step 1.4

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

$e^{\frac{x^{2}}{2}}$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

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

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Step 6.1.1

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

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Step 6.1.1.1

Differentiate $\frac{x^{2}}{2}$ .

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

Step 6.1.1.2

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

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

Step 6.1.1.3

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

$\frac{1}{2} (2 x)$

Step 6.1.1.4

Simplify terms.

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Step 6.1.1.4.1

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

$\frac{2}{2} x$

Step 6.1.1.4.2

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

$\frac{2 x}{2}$

Step 6.1.1.4.3

Cancel the common factor of $2$ .

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Step 6.1.1.4.3.1

Cancel the common factor.

$\frac{2 x}{2}$

Step 6.1.1.4.3.2

Divide $x$ by $1$ .

$x$

Step 6.1.2

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

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

Step 6.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = e^{u_{1}}$ and $d v = \sqrt{2 u_{1}}$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Step 6.3.2

Multiply $2$ by $3$ .

$e^{\frac{x^{2}}{2}} y = e^{u_{1}} (\frac{2}{6} {u_{2}}^{\frac{3}{2}}) - \int \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1}$

Step 6.3.3

Cancel the common factor of $2$ and $6$ .

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Step 6.3.3.1

Factor $2$ out of $2$ .

$e^{\frac{x^{2}}{2}} y = e^{u_{1}} (\frac{2 (1)}{6} {u_{2}}^{\frac{3}{2}}) - \int \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1}$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $6$ .

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

Step 6.3.3.2.2

Cancel the common factor.

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

Step 6.3.3.2.3

Rewrite the expression.

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

Step 6.3.4

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

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

Step 6.3.5

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

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

Step 6.4

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

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

Step 6.5

Simplify.

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Step 6.5.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Step 6.5.2

Multiply $2$ by $3$ .

$e^{\frac{x^{2}}{2}} y = \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - (\frac{2}{6} {u_{2}}^{\frac{3}{2}} \int e^{u_{1}} d u_{1})$

Step 6.5.3

Cancel the common factor of $2$ and $6$ .

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Step 6.5.3.1

Factor $2$ out of $2$ .

$e^{\frac{x^{2}}{2}} y = \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - (\frac{2 (1)}{6} {u_{2}}^{\frac{3}{2}} \int e^{u_{1}} d u_{1})$

Step 6.5.3.2

Cancel the common factors.

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Step 6.5.3.2.1

Factor $2$ out of $6$ .

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

Step 6.5.3.2.2

Cancel the common factor.

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

Step 6.5.3.2.3

Rewrite the expression.

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

Step 6.5.4

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

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

Step 6.6

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

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

Step 6.7

Simplify.

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Step 6.7.1

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

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

Step 6.7.2

Simplify.

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Step 6.7.2.1

Subtract $\frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3}$ from $\frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3}$ .

$e^{\frac{x^{2}}{2}} y = 0 + C$

Step 6.7.2.2

Add $0$ and $C$ .

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

Step 7

Divide each term in $e^{\frac{x^{2}}{2}} y = C$ by $e^{\frac{x^{2}}{2}}$ and simplify.

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Step 7.1

Divide each term in $e^{\frac{x^{2}}{2}} y = C$ by $e^{\frac{x^{2}}{2}}$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $0$ for $y$ in $y = \frac{C}{e^{\frac{x^{2}}{2}}}$ .

$0 = \frac{C}{e^{\frac{0^{2}}{2}}}$

Step 9

Set the numerator equal to zero.

$C = 0$

Step 10

Substitute $0$ for $C$ in $y = \frac{C}{e^{\frac{x^{2}}{2}}}$ and simplify.

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Step 10.1

Substitute $0$ for $C$ .

$y = \frac{0}{e^{\frac{x^{2}}{2}}}$

Step 10.2

Divide $0$ by $e^{\frac{x^{2}}{2}}$ .

$y = 0$