Calculus Examples

$\frac{d y}{d x} + \frac{y}{x} = 5 x^{3}$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Factor $y$ out of $\frac{y}{x}$ .

$\frac{d y}{d x} + y \frac{1}{x} = 5 x^{3}$

Step 1.2

Reorder $y$ and $\frac{1}{x}$ .

$\frac{d y}{d x} + \frac{1}{x} y = 5 x^{3}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x}$ .

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Step 2.1

Set up the integration.

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

Step 2.2

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 2.4

Exponentiation and log are inverse functions.

$x$

Step 3

Multiply each term by the integrating factor $x$ .

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Step 3.1

Multiply each term by $x$ .

$x \frac{d y}{d x} + x (\frac{1}{x} y) = x (5 x^{3})$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{x}$ and $y$ .

$x \frac{d y}{d x} + x \frac{y}{x} = x (5 x^{3})$

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

Cancel the common factor.

$x \frac{d y}{d x} + x \frac{y}{x} = x (5 x^{3})$

Step 3.2.2.2

Rewrite the expression.

$x \frac{d y}{d x} + y = x (5 x^{3})$

Step 3.3

Rewrite using the commutative property of multiplication.

$x \frac{d y}{d x} + y = 5 x \cdot x^{3}$

Step 3.4

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.4.1

Move $x^{3}$ .

$x \frac{d y}{d x} + y = 5 (x^{3} x)$

Step 3.4.2

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

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Step 3.4.2.1

Raise $x$ to the power of $1$ .

$x \frac{d y}{d x} + y = 5 (x^{3} x^{1})$

Step 3.4.2.2

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

$x \frac{d y}{d x} + y = 5 x^{3 + 1}$

Step 3.4.3

Add $3$ and $1$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify the answer.

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Step 7.3.1

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

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

Step 7.3.2

Simplify.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor of $5$ .

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Step 7.3.2.2.1

Cancel the common factor.

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

Step 7.3.2.2.2

Rewrite the expression.

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

Step 7.3.2.3

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

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

$y = \frac{x \cdot x^{4}}{x} + \frac{C}{x}$

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Raise $x$ to the power of $1$ .

$y = \frac{x \cdot x^{4}}{x^{1}} + \frac{C}{x}$

Step 8.3.1.2.2

Factor $x$ out of $x^{1}$ .

$y = \frac{x \cdot x^{4}}{x \cdot 1} + \frac{C}{x}$

Step 8.3.1.2.3

Cancel the common factor.

$y = \frac{x \cdot x^{4}}{x \cdot 1} + \frac{C}{x}$

Step 8.3.1.2.4

Rewrite the expression.

$y = \frac{x^{4}}{1} + \frac{C}{x}$

Step 8.3.1.2.5

Divide $x^{4}$ by $1$ .

$y = x^{4} + \frac{C}{x}$