Calculus Examples

$\frac{d y}{d x} = 2 x y + 4 x^{2}$

Step 1

Subtract $2 x y$ from both sides of the equation.

$\frac{d y}{d x} - 2 x y = 4 x^{2}$

Step 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 2.1

Set up the integration.

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

Step 2.2

Integrate $- 2 x$ .

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Step 2.2.1

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

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

Step 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 2.2.3

Simplify the answer.

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Step 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 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

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

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Step 2.2.3.2.2.1

Factor $2$ out of $- 2$ .

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

Step 2.2.3.2.2.2

Cancel the common factors.

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Step 2.2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.3.2.2.2.2

Cancel the common factor.

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

Step 2.2.3.2.2.2.3

Rewrite the expression.

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

Step 2.2.3.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 2.3

Remove the constant of integration.

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

Step 3

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

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Step 3.1

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

$e^{- x^{2}} \frac{d y}{d x} + e^{- x^{2}} (- 2 x y) = e^{- x^{2}} (4 x^{2})$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = e^{- x^{2}} (4 x^{2})$

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 4 e^{- x^{2}} x^{2}$

Step 3.4

Reorder factors in $e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 4 e^{- x^{2}} x^{2}$ .

$e^{- x^{2}} \frac{d y}{d x} - 2 x y e^{- x^{2}} = 4 x^{2} e^{- x^{2}}$

Step 4

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

$\frac{d}{d x} [e^{- x^{2}} y] = 4 x^{2} e^{- x^{2}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- x^{2}} y] d x = \int 4 x^{2} e^{- x^{2}} d x$

Step 6

Integrate the left side.

$e^{- x^{2}} y = \int 4 x^{2} e^{- x^{2}} d x$

Step 7

Integrate the right side.

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Step 7.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$e^{- x^{2}} y = 4 \int x^{2} e^{- x^{2}} d x$

Step 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 7.2.1

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

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Step 7.2.1.1

Differentiate $- x^{2}$ .

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

Step 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 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 7.2.1.4

Multiply $2$ by $- 1$ .

$- 2 x$

Step 7.2.2

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

$e^{- x^{2}} y = 4 \int \sqrt{- u_{1}} e^{u_{1}} \frac{1}{- 2} d u_{1}$

Step 7.3

Simplify.

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Step 7.3.1

Move the negative in front of the fraction.

$e^{- x^{2}} y = 4 \int \sqrt{- u_{1}} e^{u_{1}} (- \frac{1}{2}) d u_{1}$

Step 7.3.2

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

$e^{- x^{2}} y = 4 \int e^{u_{1}} (- \frac{\sqrt{- u_{1}}}{2}) d u_{1}$

Step 7.3.3

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

$e^{- x^{2}} y = 4 \int - \frac{e^{u_{1}} \sqrt{- u_{1}}}{2} d u_{1}$

Step 7.4

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

$e^{- x^{2}} y = 4 (- \int \frac{e^{u_{1}} \sqrt{- u_{1}}}{2} d u_{1})$

Step 7.5

Multiply $- 1$ by $4$ .

$e^{- x^{2}} y = - 4 \int \frac{e^{u_{1}} \sqrt{- u_{1}}}{2} d u_{1}$

Step 7.6

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

$e^{- x^{2}} y = - 4 (\frac{1}{2} \int e^{u_{1}} \sqrt{- u_{1}} d u_{1})$

Step 7.7

Simplify.

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Step 7.7.1

Combine $\frac{1}{2}$ and $- 4$ .

$e^{- x^{2}} y = \frac{- 4}{2} \int e^{u_{1}} \sqrt{- u_{1}} d u_{1}$

Step 7.7.2

Cancel the common factor of $- 4$ and $2$ .

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Step 7.7.2.1

Factor $2$ out of $- 4$ .

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

Step 7.7.2.2

Cancel the common factors.

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Step 7.7.2.2.1

Factor $2$ out of $2$ .

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

Step 7.7.2.2.2

Cancel the common factor.

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

Step 7.7.2.2.3

Rewrite the expression.

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

Step 7.7.2.2.4

Divide $- 2$ by $1$ .

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

Step 7.8

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

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

Step 7.9

Simplify.

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Step 7.9.1

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

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

Step 7.9.2

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

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

Step 7.9.3

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

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

Step 7.9.4

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

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

Step 7.9.5

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

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

Step 7.9.6

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

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

Step 7.9.7

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

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

Step 7.9.8

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

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

Step 7.10

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

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

Step 7.11

Simplify.

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Step 7.11.1

Multiply $- 1$ by $- 1$ .

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

Step 7.11.2

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

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

Step 7.12

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.

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

Step 7.13

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

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

Step 7.14

Simplify.

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Step 7.14.1

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

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

Step 7.14.2

Simplify.

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Step 7.14.2.1

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

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

Step 7.14.2.2

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

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

Step 7.14.2.3

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

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

Step 7.14.2.4

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

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

Step 7.14.2.5

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

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

Step 7.14.2.6

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

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

Step 7.14.2.7

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

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

Step 7.14.2.8

Multiply $- 2$ by $0$ .

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

Step 7.14.2.9

Add $0$ and $C$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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