Calculus Examples

$\frac{d y}{d x} + (a x + b) y = 0$

Step 1

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

Tap for more steps...

Step 1.1

Set up the integration.

$e^{\int a x + b d x}$

Step 1.2

Integrate $a x + b$ .

Tap for more steps...

Step 1.2.1

Split the single integral into multiple integrals.

$e^{\int a x d x + \int b d x}$

Step 1.2.2

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

$e^{a \int x d x + \int b d x}$

Step 1.2.3

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

$e^{a (\frac{1}{2} x^{2} + C) + \int b d x}$

Step 1.2.4

Apply the constant rule.

$e^{a (\frac{1}{2} x^{2} + C) + b x + C}$

Step 1.2.5

Simplify.

Tap for more steps...

Step 1.2.5.1

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

$e^{a (\frac{x^{2}}{2} + C) + b x + C}$

Step 1.2.5.2

Simplify.

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

Step 1.3

Remove the constant of integration.

$e^{\frac{1}{2} a x^{2} + b x}$

Step 1.4

Simplify each term.

Tap for more steps...

Step 1.4.1

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

$e^{\frac{a}{2} x^{2} + b x}$

Step 1.4.2

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

$e^{\frac{a x^{2}}{2} + b x}$

Step 2

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

Tap for more steps...

Step 2.1

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

$e^{\frac{a x^{2}}{2} + b x} \frac{d y}{d x} + e^{\frac{a x^{2}}{2} + b x} ((a x + b) y) = e^{\frac{a x^{2}}{2} + b x} \cdot 0$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Apply the distributive property.

$e^{\frac{a x^{2}}{2} + b x} \frac{d y}{d x} + e^{\frac{a x^{2}}{2} + b x} (a x y + b y) = e^{\frac{a x^{2}}{2} + b x} \cdot 0$

Step 2.2.2

Apply the distributive property.

$e^{\frac{a x^{2}}{2} + b x} \frac{d y}{d x} + e^{\frac{a x^{2}}{2} + b x} (a x y) + e^{\frac{a x^{2}}{2} + b x} (b y) = e^{\frac{a x^{2}}{2} + b x} \cdot 0$

Step 2.3

Multiply $e^{\frac{a x^{2}}{2} + b x}$ by $0$ .

$e^{\frac{a x^{2}}{2} + b x} \frac{d y}{d x} + e^{\frac{a x^{2}}{2} + b x} (a x y) + e^{\frac{a x^{2}}{2} + b x} (b y) = 0$

Step 2.4

Reorder factors in $e^{\frac{a x^{2}}{2} + b x} \frac{d y}{d x} + e^{\frac{a x^{2}}{2} + b x} (a x y) + e^{\frac{a x^{2}}{2} + b x} (b y) = 0$ .

$e^{\frac{a x^{2}}{2} + b x} \frac{d y}{d x} + a x y e^{\frac{a x^{2}}{2} + b x} + b y e^{\frac{a x^{2}}{2} + b x} = 0$

Step 3

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

$\frac{d}{d x} [e^{\frac{a x^{2}}{2} + b x} y] = 0$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{\frac{a x^{2}}{2} + b x} y] d x = \int 0 d x$

Step 5

Integrate the left side.

$e^{\frac{a x^{2}}{2} + b x} y = \int 0 d x$

Step 6

Integrate the right side.

Tap for more steps...

Step 6.1

The integral of $0$ with respect to $x$ is $0$ .

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

Step 6.2

Add $0$ and $C$ .

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

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

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

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Simplify the denominator.

Tap for more steps...

Step 7.3.1.1

To write $b x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{C}{e^{\frac{a x^{2}}{2} + b x \cdot \frac{2}{2}}}$

Step 7.3.1.2

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

$y = \frac{C}{e^{\frac{a x^{2}}{2} + \frac{b x \cdot 2}{2}}}$

Step 7.3.1.3

Combine the numerators over the common denominator.

$y = \frac{C}{e^{\frac{a x^{2} + b x \cdot 2}{2}}}$

Step 7.3.1.4

Simplify the numerator.

Tap for more steps...

Step 7.3.1.4.1

Factor $x$ out of $a x^{2} + b x \cdot 2$ .

Tap for more steps...

Step 7.3.1.4.1.1

Factor $x$ out of $a x^{2}$ .

$y = \frac{C}{e^{\frac{x (a x) + b x \cdot 2}{2}}}$

Step 7.3.1.4.1.2

Factor $x$ out of $b x \cdot 2$ .

$y = \frac{C}{e^{\frac{x (a x) + x (b \cdot 2)}{2}}}$

Step 7.3.1.4.1.3

Factor $x$ out of $x (a x) + x (b \cdot 2)$ .

$y = \frac{C}{e^{\frac{x (a x + b \cdot 2)}{2}}}$

Step 7.3.1.4.2

Move $2$ to the left of $b$ .

$y = \frac{C}{e^{\frac{x (a x + 2 b)}{2}}}$