Calculus Examples

$\frac{d y}{d x} - \frac{y}{x} = x e^{x}$ , $y (1) = e - 1$

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 $- \frac{y}{x}$ .

$\frac{d y}{d x} + y (- \frac{1}{x}) = x e^{x}$

Step 1.2

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

$\frac{d y}{d x} - \frac{1}{x} y = x e^{x}$

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

Integrate $- \frac{1}{x}$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 1})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 2.6

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

$\frac{1}{x}$

Step 3

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

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Step 3.1

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

$\frac{1}{x} \frac{d y}{d x} + \frac{1}{x} (- \frac{1}{x} y) = \frac{1}{x} (x e^{x})$

Step 3.2

Simplify each term.

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Step 3.2.1

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

$\frac{\frac{d y}{d x}}{x} + \frac{1}{x} (- \frac{1}{x} y) = \frac{1}{x} (x e^{x})$

Step 3.2.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x} - \frac{1}{x} (\frac{1}{x} y) = \frac{1}{x} (x e^{x})$

Step 3.2.3

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

$\frac{\frac{d y}{d x}}{x} - \frac{1}{x} \cdot \frac{y}{x} = \frac{1}{x} (x e^{x})$

Step 3.2.4

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

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Step 3.2.4.1

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

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x \cdot x} = \frac{1}{x} (x e^{x})$

Step 3.2.4.2

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

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x^{1} x} = \frac{1}{x} (x e^{x})$

Step 3.2.4.3

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

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x^{1} x^{1}} = \frac{1}{x} (x e^{x})$

Step 3.2.4.4

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

$\frac{\frac{d y}{d x}}{x} - \frac{y}{x^{1 + 1}} = \frac{1}{x} (x e^{x})$

Step 3.2.4.5

Add $1$ and $1$ .

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

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

$\frac{d}{d x} [\frac{1}{x} y] = e^{x}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} y] d x = \int e^{x} d x$

Step 6

Integrate the left side.

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

Step 7

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

Multiply both sides by $x$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Cancel the common factor of $x$ .

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Rewrite the expression.

$y = (e^{x} + C) x$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(e^{x} + C) x$ .

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Step 8.3.2.1.1

Apply the distributive property.

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

Step 8.3.2.1.2

Simplify the expression.

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Step 8.3.2.1.2.1

Reorder factors in $e^{x} x + C x$ .

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

Step 8.3.2.1.2.2

Reorder $x e^{x}$ and $C x$ .

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

Step 9

Use the initial condition to find the value of $C$ by substituting $1$ for $x$ and $e - 1$ for $y$ in $y = C x + x e^{x}$ .

$e - 1 = C \cdot 1 + 1 e^{1}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $C \cdot 1 + 1 e^{1} = e - 1$ .

$C \cdot 1 + 1 e = e - 1$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $C$ by $1$ .

$C + 1 e^{1} = e - 1$

Step 10.2.2

Multiply $e^{1}$ by $1$ .

$C + e^{1} = e - 1$

Step 10.2.3

Simplify.

$C + e = e - 1$

Step 10.3

Move all terms not containing $C$ to the right side of the equation.

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Step 10.3.1

Subtract $e$ from both sides of the equation.

$C = e - 1 - e$

Step 10.3.2

Subtract $e$ from $e$ .

$C = 0 - 1$

Step 10.3.3

Subtract $1$ from $0$ .

$C = - 1$

Step 11

Substitute $- 1$ for $C$ in $y = C x + x e^{x}$ and simplify.

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Step 11.1

Substitute $- 1$ for $C$ .

$y = - 1 x + x e^{x}$

Step 11.2

Rewrite $- 1 x$ as $- x$ .

$y = - x + x e^{x}$