Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 3 x^{2}$ .

Tap for more steps...

Step 1.1

Set up the integration.

$e^{\int 3 x^{2} d x}$

Step 1.2

Integrate $3 x^{2}$ .

Tap for more steps...

Step 1.2.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{3 \int x^{2} d x}$

Step 1.2.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$e^{3 (\frac{1}{3} x^{3} + C)}$

Step 1.2.3

Simplify the answer.

Tap for more steps...

Step 1.2.3.1

Rewrite $3 (\frac{1}{3} x^{3} + C)$ as $3 (\frac{1}{3}) x^{3} + C$ .

$e^{3 (\frac{1}{3}) x^{3} + C}$

Step 1.2.3.2

Simplify.

Tap for more steps...

Step 1.2.3.2.1

Combine $3$ and $\frac{1}{3}$ .

$e^{\frac{3}{3} x^{3} + C}$

Step 1.2.3.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.3.2.2.1

Cancel the common factor.

$e^{\frac{3}{3} x^{3} + C}$

Step 1.2.3.2.2.2

Rewrite the expression.

$e^{1 x^{3} + C}$

Step 1.2.3.2.3

Multiply $x^{3}$ by $1$ .

$e^{x^{3} + C}$

Step 1.3

Remove the constant of integration.

$e^{x^{3}}$

Step 2

Multiply each term by the integrating factor $e^{x^{3}}$ .

Tap for more steps...

Step 2.1

Multiply each term by $e^{x^{3}}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

Tap for more steps...

Step 6.1

Let $u_{2} = e^{x^{3}}$ . Then $d u_{2} = 3 x^{2} e^{x^{3}} d x$ , so $\frac{1}{3} d u_{2} = x^{2} e^{x^{3}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 6.1.1

Let $u_{2} = e^{x^{3}}$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 6.1.1.1

Differentiate $e^{x^{3}}$ .

$\frac{d}{d x} [e^{x^{3}}]$

Step 6.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{3}$ .

Tap for more steps...

Step 6.1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x^{3}]$

Step 6.1.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [x^{3}]$

Step 6.1.1.2.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$e^{x^{3}} \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$e^{x^{3}} (3 x^{2})$

Step 6.1.1.4

Simplify.

Tap for more steps...

Step 6.1.1.4.1

Reorder the factors of $e^{x^{3}} (3 x^{2})$ .

$3 e^{x^{3}} x^{2}$

Step 6.1.1.4.2

Reorder factors in $3 e^{x^{3}} x^{2}$ .

$3 x^{2} e^{x^{3}}$

Step 6.1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 6.2

Apply the constant rule.

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

Step 6.3

Replace all occurrences of $u_{2}$ with $e^{x^{3}}$ .

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

Step 7

Divide each term in $e^{x^{3}} y = \frac{1}{3} e^{x^{3}} + C$ by $e^{x^{3}}$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $e^{x^{3}} y = \frac{1}{3} e^{x^{3}} + C$ by $e^{x^{3}}$ .

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

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $e^{x^{3}}$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Cancel the common factor of $e^{x^{3}}$ .

Tap for more steps...

Step 7.3.1.1

Cancel the common factor.

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

Step 7.3.1.2

Divide $\frac{1}{3}$ by $1$ .

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