Calculus Examples

$\frac{d y}{d x} + 2 x y = - 2 x^{3}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 2 x$ .

Tap for more steps...

Step 1.1

Set up the integration.

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

Step 1.2

Integrate $2 x$ .

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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 1.2.3

Simplify the answer.

Tap for more steps...

Step 1.2.3.1

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 1.2.3.2

Simplify.

Tap for more steps...

Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.2.1

Cancel the common factor.

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

Step 1.2.3.2.2.2

Rewrite the expression.

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

Step 1.2.3.2.3

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

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

Step 1.3

Remove the constant of integration.

$e^{x^{2}}$

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 2.4

Reorder factors in $e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = - 2 e^{x^{2}} x^{3}$ .

$e^{x^{2}} \frac{d y}{d x} + 2 x y e^{x^{2}} = - 2 x^{3} e^{x^{2}}$

Step 3

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

$\frac{d}{d x} [e^{x^{2}} y] = - 2 x^{3} e^{x^{2}}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x^{2}} y] d x = \int - 2 x^{3} e^{x^{2}} d x$

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

Tap for more steps...

Step 6.2.1.1

Differentiate $x^{2}$ .

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

Step 6.2.1.2

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 6.2.2

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

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

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.4

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

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

Step 6.5

Simplify.

Tap for more steps...

Step 6.5.1

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

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

Step 6.5.2

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

Tap for more steps...

Step 6.5.2.1

Factor $2$ out of $- 2$ .

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

Step 6.5.2.2

Cancel the common factors.

Tap for more steps...

Step 6.5.2.2.1

Factor $2$ out of $2$ .

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

Step 6.5.2.2.2

Cancel the common factor.

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

Step 6.5.2.2.3

Rewrite the expression.

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

Step 6.5.2.2.4

Divide $- 1$ by $1$ .

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

Step 6.6

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

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

Step 6.7

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

$e^{x^{2}} y = - (u e^{u} - (e^{u} + C))$

Step 6.8

Simplify.

$e^{x^{2}} y = - (u e^{u} - e^{u}) + C$

Step 6.9

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

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

Step 6.10

Simplify.

Tap for more steps...

Step 6.10.1

Apply the distributive property.

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

Step 6.10.2

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

Tap for more steps...

Step 6.10.2.1

Multiply $- 1$ by $- 1$ .

$e^{x^{2}} y = - x^{2} e^{x^{2}} + 1 e^{x^{2}} + C$

Step 6.10.2.2

Multiply $e^{x^{2}}$ by $1$ .

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

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

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

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Simplify each term.

Tap for more steps...

Step 7.3.1.1

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

Tap for more steps...

Step 7.3.1.1.1

Cancel the common factor.

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

Step 7.3.1.1.2

Divide $- x^{2}$ by $1$ .

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

Step 7.3.1.2

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

Tap for more steps...

Step 7.3.1.2.1

Cancel the common factor.

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

Step 7.3.1.2.2

Rewrite the expression.

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