Calculus Examples

$\frac{d y}{d x} + \frac{2}{x} y = 5 x^{2}$

Step 1

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

Tap for more steps...

Step 1.1

Set up the integration.

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

Step 1.2

Integrate $\frac{2}{x}$ .

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$e^{2 (\ln (| x |) + C)}$

Step 1.2.3

Simplify.

$e^{2 \ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{2 \ln (x)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (x^{2})}$

Step 1.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 2

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

Tap for more steps...

Step 2.1

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

$x^{2} \frac{d y}{d x} + x^{2} (\frac{2}{x} y) = x^{2} (5 x^{2})$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

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

$x^{2} \frac{d y}{d x} + x^{2} \frac{2 y}{x} = x^{2} (5 x^{2})$

Step 2.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.2.1

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

$x^{2} \frac{d y}{d x} + x \cdot x \frac{2 y}{x} = x^{2} (5 x^{2})$

Step 2.2.2.2

Cancel the common factor.

$x^{2} \frac{d y}{d x} + x \cdot x \frac{2 y}{x} = x^{2} (5 x^{2})$

Step 2.2.2.3

Rewrite the expression.

$x^{2} \frac{d y}{d x} + x (2 y) = x^{2} (5 x^{2})$

Step 2.2.3

Rewrite using the commutative property of multiplication.

$x^{2} \frac{d y}{d x} + 2 x y = x^{2} (5 x^{2})$

Step 2.3

Rewrite using the commutative property of multiplication.

$x^{2} \frac{d y}{d x} + 2 x y = 5 x^{2} x^{2}$

Step 2.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 2.4.1

Move $x^{2}$ .

$x^{2} \frac{d y}{d x} + 2 x y = 5 (x^{2} x^{2})$

Step 2.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} \frac{d y}{d x} + 2 x y = 5 x^{2 + 2}$

Step 2.4.3

Add $2$ and $2$ .

$x^{2} \frac{d y}{d x} + 2 x y = 5 x^{4}$

Step 3

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

$\frac{d}{d x} [x^{2} y] = 5 x^{4}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int 5 x^{4} d x$

Step 5

Integrate the left side.

$x^{2} y = \int 5 x^{4} d x$

Step 6

Integrate the right side.

Tap for more steps...

Step 6.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$x^{2} y = 5 \int x^{4} d x$

Step 6.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$x^{2} y = 5 (\frac{1}{5} x^{5} + C)$

Step 6.3

Simplify the answer.

Tap for more steps...

Step 6.3.1

Rewrite $5 (\frac{1}{5} x^{5} + C)$ as $5 (\frac{1}{5}) x^{5} + C$ .

$x^{2} y = 5 (\frac{1}{5}) x^{5} + C$

Step 6.3.2

Simplify.

Tap for more steps...

Step 6.3.2.1

Combine $5$ and $\frac{1}{5}$ .

$x^{2} y = \frac{5}{5} x^{5} + C$

Step 6.3.2.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 6.3.2.2.1

Cancel the common factor.

$x^{2} y = \frac{5}{5} x^{5} + C$

Step 6.3.2.2.2

Rewrite the expression.

$x^{2} y = 1 x^{5} + C$

Step 6.3.2.3

Multiply $x^{5}$ by $1$ .

$x^{2} y = x^{5} + C$

Step 7

Divide each term in $x^{2} y = x^{5} + C$ by $x^{2}$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $x^{2} y = x^{5} + C$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{x^{5}}{x^{2}} + \frac{C}{x^{2}}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

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

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{x^{5}}{x^{2}} + \frac{C}{x^{2}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{5}}{x^{2}} + \frac{C}{x^{2}}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Cancel the common factor of $x^{5}$ and $x^{2}$ .

Tap for more steps...

Step 7.3.1.1

Factor $x^{2}$ out of $x^{5}$ .

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

Step 7.3.1.2

Cancel the common factors.

Tap for more steps...

Step 7.3.1.2.1

Multiply by $1$ .

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

Step 7.3.1.2.2

Cancel the common factor.

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

Step 7.3.1.2.3

Rewrite the expression.

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

Step 7.3.1.2.4

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

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