Calculus Examples

$x (\frac{d y}{d x}) - 2 y = x^{3} \cos (4 x)$

Step 1

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

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Step 1.1

Divide each term in $x \frac{d y}{d x} - 2 y = x^{3} \cos (4 x)$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} + \frac{- 2 y}{x} = \frac{x^{3} \cos (4 x)}{x}$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} + \frac{- 2 y}{x} = \frac{x^{3} \cos (4 x)}{x}$

Step 1.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} + \frac{- 2 y}{x} = \frac{x^{3} \cos (4 x)}{x}$

Step 1.3

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

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Step 1.3.1

Factor $x$ out of $x^{3} \cos (4 x)$ .

$\frac{d y}{d x} + \frac{- 2 y}{x} = \frac{x (x^{2} \cos (4 x))}{x}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{d y}{d x} + \frac{- 2 y}{x} = \frac{x (x^{2} \cos (4 x))}{x^{1}}$

Step 1.3.2.2

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

$\frac{d y}{d x} + \frac{- 2 y}{x} = \frac{x (x^{2} \cos (4 x))}{x \cdot 1}$

Step 1.3.2.3

Cancel the common factor.

$\frac{d y}{d x} + \frac{- 2 y}{x} = \frac{x (x^{2} \cos (4 x))}{x \cdot 1}$

Step 1.3.2.4

Rewrite the expression.

$\frac{d y}{d x} + \frac{- 2 y}{x} = \frac{x^{2} \cos (4 x)}{1}$

Step 1.3.2.5

Divide $x^{2} \cos (4 x)$ by $1$ .

$\frac{d y}{d x} + \frac{- 2 y}{x} = x^{2} \cos (4 x)$

Step 1.4

Factor $y$ out of $\frac{- 2 y}{x}$ .

$\frac{d y}{d x} + y \frac{- 2}{x} = x^{2} \cos (4 x)$

Step 1.5

Reorder $y$ and $\frac{- 2}{x}$ .

$\frac{d y}{d x} + \frac{- 2}{x} y = x^{2} \cos (4 x)$

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $2$ by $- 1$ .

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 2}$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{2}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{2}}$ .

$\frac{1}{x^{2}} \frac{d y}{d x} + \frac{1}{x^{2}} (\frac{- 2}{x} y) = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2

Simplify each term.

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Step 3.2.1

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

$\frac{\frac{d y}{d x}}{x^{2}} + \frac{1}{x^{2}} (\frac{- 2}{x} y) = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2.2

Move the negative in front of the fraction.

$\frac{\frac{d y}{d x}}{x^{2}} + \frac{1}{x^{2}} (- \frac{2}{x} y) = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{1}{x^{2}} (\frac{2}{x} y) = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2.4

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

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{1}{x^{2}} \cdot \frac{2 y}{x} = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2.5

Multiply $- \frac{1}{x^{2}} \cdot \frac{2 y}{x}$ .

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Step 3.2.5.1

Multiply $\frac{2 y}{x}$ by $\frac{1}{x^{2}}$ .

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x \cdot x^{2}} = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2.5.2

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

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Step 3.2.5.2.1

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

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Step 3.2.5.2.1.1

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

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{1} x^{2}} = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2.5.2.1.2

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

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{1 + 2}} = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.2.5.2.2

Add $1$ and $2$ .

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{3}} = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.3

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

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Step 3.3.1

Factor $x^{2}$ out of $x^{2} \cos (4 x)$ .

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{3}} = \frac{1}{x^{2}} (x^{2} (\cos (4 x)))$

Step 3.3.2

Cancel the common factor.

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{3}} = \frac{1}{x^{2}} (x^{2} \cos (4 x))$

Step 3.3.3

Rewrite the expression.

$\frac{\frac{d y}{d x}}{x^{2}} - \frac{2 y}{x^{3}} = \cos (4 x)$

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{2}} y] = \cos (4 x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{2}} y] d x = \int \cos (4 x) d x$

Step 6

Integrate the left side.

$\frac{1}{x^{2}} y = \int \cos (4 x) d x$

Step 7

Integrate the right side.

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Step 7.1

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 7.1.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 7.1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 7.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 \cdot 1$

Step 7.1.1.4

Multiply $4$ by $1$ .

$4$

Step 7.1.2

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

$\frac{1}{x^{2}} y = \int \cos (u) \frac{1}{4} d u$

Step 7.2

Combine $\cos (u)$ and $\frac{1}{4}$ .

$\frac{1}{x^{2}} y = \int \frac{\cos (u)}{4} d u$

Step 7.3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{x^{2}} y = \frac{1}{4} \int \cos (u) d u$

Step 7.4

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{1}{x^{2}} y = \frac{1}{4} (\sin (u) + C)$

Step 7.5

Simplify.

$\frac{1}{x^{2}} y = \frac{1}{4} \sin (u) + C$

Step 7.6

Replace all occurrences of $u$ with $4 x$ .

$\frac{1}{x^{2}} y = \frac{1}{4} \sin (4 x) + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify the left side.

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Step 8.1.1

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

$\frac{y}{x^{2}} = \frac{1}{4} \sin (4 x) + C$

Step 8.2

Simplify the right side.

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Step 8.2.1

Combine $\frac{1}{4}$ and $\sin (4 x)$ .

$\frac{y}{x^{2}} = \frac{\sin (4 x)}{4} + C$

Step 8.3

Multiply both sides by $x^{2}$ .

$\frac{y}{x^{2}} x^{2} = (\frac{\sin (4 x)}{4} + C) x^{2}$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

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

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Step 8.4.1.1.1

Cancel the common factor.

$\frac{y}{x^{2}} x^{2} = (\frac{\sin (4 x)}{4} + C) x^{2}$

Step 8.4.1.1.2

Rewrite the expression.

$y = (\frac{\sin (4 x)}{4} + C) x^{2}$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\frac{\sin (4 x)}{4} + C) x^{2}$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = \frac{\sin (4 x)}{4} x^{2} + C x^{2}$

Step 8.4.2.1.2

Combine $\frac{\sin (4 x)}{4}$ and $x^{2}$ .

$y = \frac{\sin (4 x) x^{2}}{4} + C x^{2}$

Step 8.4.2.1.3

Simplify the expression.

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Step 8.4.2.1.3.1

Reorder factors in $\frac{\sin (4 x) x^{2}}{4} + C x^{2}$ .

$y = \frac{x^{2} \sin (4 x)}{4} + C x^{2}$

Step 8.4.2.1.3.2

Reorder $\frac{x^{2} \sin (4 x)}{4}$ and $C x^{2}$ .

$y = C x^{2} + \frac{x^{2} \sin (4 x)}{4}$