Calculus Examples

$\frac{d y}{d x} = x + 5 y$ , $y (0) = 3$

Step 1

Subtract $5 y$ from both sides of the equation.

$\frac{d y}{d x} - 5 y = x$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 5$ .

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Step 2.1

Set up the integration.

$e^{\int - 5 d x}$

Step 2.2

Apply the constant rule.

$e^{- 5 x + C}$

Step 2.3

Remove the constant of integration.

$e^{- 5 x}$

Step 3

Multiply each term by the integrating factor $e^{- 5 x}$ .

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Step 3.1

Multiply each term by $e^{- 5 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- 5 x} \frac{d y}{d x} - 5 e^{- 5 x} y = e^{- 5 x} x$

Step 3.3

Reorder factors in $e^{- 5 x} \frac{d y}{d x} - 5 e^{- 5 x} y = e^{- 5 x} x$ .

$e^{- 5 x} \frac{d y}{d x} - 5 y e^{- 5 x} = x e^{- 5 x}$

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

$e^{- 5 x} y = \int x e^{- 5 x} d x$

Step 7

Integrate the right side.

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Step 7.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 5 x}$ .

$e^{- 5 x} y = x (- \frac{1}{5} e^{- 5 x}) - \int - \frac{1}{5} e^{- 5 x} d x$

Step 7.2

Simplify.

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Step 7.2.1

Combine $e^{- 5 x}$ and $\frac{1}{5}$ .

$e^{- 5 x} y = x (- \frac{e^{- 5 x}}{5}) - \int - \frac{1}{5} e^{- 5 x} d x$

Step 7.2.2

Combine $x$ and $\frac{e^{- 5 x}}{5}$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \int - \frac{1}{5} e^{- 5 x} d x$

Step 7.3

Since $- \frac{1}{5}$ is constant with respect to $x$ , move $- \frac{1}{5}$ out of the integral.

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - (- \frac{1}{5} \int e^{- 5 x} d x)$

Step 7.4

Simplify.

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} + 1 (\frac{1}{5} \int e^{- 5 x} d x)$

Step 7.4.2

Multiply $\frac{1}{5}$ by $1$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int e^{- 5 x} d x$

Step 7.5

Let $u = - 5 x$ . Then $d u = - 5 d x$ , so $- \frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.5.1

Let $u = - 5 x$ . Find $\frac{d u}{d x}$ .

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Step 7.5.1.1

Differentiate $- 5 x$ .

$\frac{d}{d x} [- 5 x]$

Step 7.5.1.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$- 5 \frac{d}{d x} [x]$

Step 7.5.1.3

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

$- 5 \cdot 1$

Step 7.5.1.4

Multiply $- 5$ by $1$ .

$- 5$

Step 7.5.2

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

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int e^{u} \frac{1}{- 5} d u$

Step 7.6

Simplify.

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Step 7.6.1

Move the negative in front of the fraction.

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int e^{u} (- \frac{1}{5}) d u$

Step 7.6.2

Combine $e^{u}$ and $\frac{1}{5}$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int - \frac{e^{u}}{5} d u$

Step 7.7

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

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} + \frac{1}{5} (- \int \frac{e^{u}}{5} d u)$

Step 7.8

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} + \frac{1}{5} (- (\frac{1}{5} \int e^{u} d u))$

Step 7.9

Simplify.

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Step 7.9.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \frac{1}{5 \cdot 5} \int e^{u} d u$

Step 7.9.2

Multiply $5$ by $5$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \frac{1}{25} \int e^{u} d u$

Step 7.10

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

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \frac{1}{25} (e^{u} + C)$

Step 7.11

Simplify.

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Step 7.11.1

Rewrite $- \frac{x e^{- 5 x}}{5} - \frac{1}{25} (e^{u} + C)$ as $- \frac{1}{5} x e^{- 5 x} - \frac{1}{25} e^{u} + C$ .

$e^{- 5 x} y = - \frac{1}{5} x e^{- 5 x} - \frac{1}{25} e^{u} + C$

Step 7.11.2

Simplify.

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Step 7.11.2.1

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

$e^{- 5 x} y = - \frac{x}{5} e^{- 5 x} - \frac{1}{25} e^{u} + C$

Step 7.11.2.2

Combine $e^{- 5 x}$ and $\frac{x}{5}$ .

$e^{- 5 x} y = - \frac{e^{- 5 x} x}{5} - \frac{1}{25} e^{u} + C$

Step 7.12

Replace all occurrences of $u$ with $- 5 x$ .

$e^{- 5 x} y = - \frac{e^{- 5 x} x}{5} - \frac{1}{25} e^{- 5 x} + C$

Step 7.13

Combine $e^{- 5 x}$ and $\frac{1}{25}$ .

$e^{- 5 x} y = - \frac{e^{- 5 x} x}{5} - \frac{e^{- 5 x}}{25} + C$

Step 7.14

Reorder terms.

$e^{- 5 x} y = - \frac{1}{5} e^{- 5 x} x - \frac{1}{25} e^{- 5 x} + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

Combine $e^{- 5 x}$ and $\frac{1}{5}$ .

$e^{- 5 x} y = - \frac{e^{- 5 x}}{5} x - \frac{1}{25} e^{- 5 x} + C$

Step 8.1.2

Combine $x$ and $\frac{e^{- 5 x}}{5}$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \frac{1}{25} e^{- 5 x} + C$

Step 8.1.3

Combine $e^{- 5 x}$ and $\frac{1}{25}$ .

$e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \frac{e^{- 5 x}}{25} + C$

Step 8.2

Divide each term in $e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \frac{e^{- 5 x}}{25} + C$ by $e^{- 5 x}$ and simplify.

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Step 8.2.1

Divide each term in $e^{- 5 x} y = - \frac{x e^{- 5 x}}{5} - \frac{e^{- 5 x}}{25} + C$ by $e^{- 5 x}$ .

$\frac{e^{- 5 x} y}{e^{- 5 x}} = \frac{- \frac{x e^{- 5 x}}{5}}{e^{- 5 x}} + \frac{- \frac{e^{- 5 x}}{25}}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{e^{- 5 x} y}{e^{- 5 x}} = \frac{- \frac{x e^{- 5 x}}{5}}{e^{- 5 x}} + \frac{- \frac{e^{- 5 x}}{25}}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{x e^{- 5 x}}{5}}{e^{- 5 x}} + \frac{- \frac{e^{- 5 x}}{25}}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{x e^{- 5 x}}{5} \cdot \frac{1}{e^{- 5 x}} + \frac{- \frac{e^{- 5 x}}{25}}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.3.1.2

Multiply $\frac{1}{e^{- 5 x}}$ by $\frac{x e^{- 5 x}}{5}$ .

$y = - \frac{x e^{- 5 x}}{e^{- 5 x} \cdot 5} + \frac{- \frac{e^{- 5 x}}{25}}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.3.1.3

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

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Step 8.2.3.1.3.1

Cancel the common factor.

$y = - \frac{x e^{- 5 x}}{e^{- 5 x} \cdot 5} + \frac{- \frac{e^{- 5 x}}{25}}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.3.1.3.2

Rewrite the expression.

$y = - \frac{x}{5} + \frac{- \frac{e^{- 5 x}}{25}}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.3.1.4

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{x}{5} - \frac{e^{- 5 x}}{25} \cdot \frac{1}{e^{- 5 x}} + \frac{C}{e^{- 5 x}}$

Step 8.2.3.1.5

Multiply $\frac{1}{e^{- 5 x}}$ by $\frac{e^{- 5 x}}{25}$ .

$y = - \frac{x}{5} - \frac{e^{- 5 x}}{e^{- 5 x} \cdot 25} + \frac{C}{e^{- 5 x}}$

Step 8.2.3.1.6

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

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Step 8.2.3.1.6.1

Cancel the common factor.

$y = - \frac{x}{5} - \frac{e^{- 5 x}}{e^{- 5 x} \cdot 25} + \frac{C}{e^{- 5 x}}$

Step 8.2.3.1.6.2

Rewrite the expression.

$y = - \frac{x}{5} - \frac{1}{25} + \frac{C}{e^{- 5 x}}$

Step 9

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $3$ for $y$ in $y = - \frac{x}{5} - \frac{1}{25} + \frac{C}{e^{- 5 x}}$ .

$3 = - \frac{0}{5} - \frac{1}{25} + \frac{C}{e^{- 5 \cdot 0}}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $- \frac{0}{5} - \frac{1}{25} + \frac{C}{e^{- 5 \cdot 0}} = 3$ .

$- \frac{0}{5} - \frac{1}{25} + \frac{C}{e^{- 5 \cdot 0}} = 3$

Step 10.2

Simplify $- \frac{0}{5} - \frac{1}{25} + \frac{C}{e^{- 5 \cdot 0}}$ .

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Step 10.2.1

Divide $0$ by $5$ .

$- 0 - \frac{1}{25} + \frac{C}{e^{- 5 \cdot 0}} = 3$

Step 10.2.2

Simplify each term.

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Step 10.2.2.1

Multiply $- 1$ by $0$ .

$0 - \frac{1}{25} + \frac{C}{e^{- 5 \cdot 0}} = 3$

Step 10.2.2.2

Simplify the denominator.

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Step 10.2.2.2.1

Multiply $- 5$ by $0$ .

$0 - \frac{1}{25} + \frac{C}{e^{0}} = 3$

Step 10.2.2.2.2

Anything raised to $0$ is $1$ .

$0 - \frac{1}{25} + \frac{C}{1} = 3$

Step 10.2.2.3

Divide $C$ by $1$ .

$0 - \frac{1}{25} + C = 3$

Step 10.2.3

Subtract $\frac{1}{25}$ from $0$ .

$- \frac{1}{25} + C = 3$

Step 10.3

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

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Step 10.3.1

Add $\frac{1}{25}$ to both sides of the equation.

$C = 3 + \frac{1}{25}$

Step 10.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$C = 3 \cdot \frac{25}{25} + \frac{1}{25}$

Step 10.3.3

Combine $3$ and $\frac{25}{25}$ .

$C = \frac{3 \cdot 25}{25} + \frac{1}{25}$

Step 10.3.4

Combine the numerators over the common denominator.

$C = \frac{3 \cdot 25 + 1}{25}$

Step 10.3.5

Simplify the numerator.

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Step 10.3.5.1

Multiply $3$ by $25$ .

$C = \frac{75 + 1}{25}$

Step 10.3.5.2

Add $75$ and $1$ .

$C = \frac{76}{25}$

Step 11

Substitute $\frac{76}{25}$ for $C$ in $y = - \frac{x}{5} - \frac{1}{25} + \frac{C}{e^{- 5 x}}$ and simplify.

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Step 11.1

Substitute $\frac{76}{25}$ for $C$ .

$y = - \frac{x}{5} - \frac{1}{25} + \frac{\frac{76}{25}}{e^{- 5 x}}$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{x}{5} - \frac{1}{25} + \frac{76}{25} \cdot \frac{1}{e^{- 5 x}}$

Step 11.2.2

Combine.

$y = - \frac{x}{5} - \frac{1}{25} + \frac{76 \cdot 1}{25 e^{- 5 x}}$

Step 11.2.3

Multiply $76$ by $1$ .

$y = - \frac{x}{5} - \frac{1}{25} + \frac{76}{25 e^{- 5 x}}$