Calculus Examples

$2 \frac{d y}{d x} + 3 y = e^{- x}$ , $y (0) = 5$

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 $2 \frac{d y}{d x} + 3 y = e^{- x}$ by $2$ .

$\frac{2 \frac{d y}{d x}}{2} + \frac{3 y}{2} = \frac{e^{- x}}{2}$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Cancel the common factor.

$\frac{2 \frac{d y}{d x}}{2} + \frac{3 y}{2} = \frac{e^{- x}}{2}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{3 y}{2} = \frac{e^{- x}}{2}$

Step 1.3

Reorder terms.

$\frac{d y}{d x} + \frac{3}{2} y = \frac{1}{2} e^{- x}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{3}{2}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{3}{2} d x}$

Step 2.2

Apply the constant rule.

$e^{\frac{3}{2} x + C}$

Step 2.3

Remove the constant of integration.

$e^{\frac{3}{2} x}$

Step 2.4

Combine $\frac{3}{2}$ and $x$ .

$e^{\frac{3 x}{2}}$

Step 3

Multiply each term by the integrating factor $e^{\frac{3 x}{2}}$ .

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Step 3.1

Multiply each term by $e^{\frac{3 x}{2}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

Combine $\frac{3}{2}$ and $e^{\frac{3 x}{2}}$ .

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

Step 3.2.3

Combine $\frac{3 e^{\frac{3 x}{2}}}{2}$ and $y$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{\frac{3 x}{2}} e^{- x}$

Step 3.4

Multiply $e^{\frac{3 x}{2}}$ by $e^{- x}$ by adding the exponents.

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Step 3.4.1

Move $e^{- x}$ .

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

Step 3.4.2

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

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{- x + \frac{3 x}{2}}$

Step 3.4.3

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{- x \cdot \frac{2}{2} + \frac{3 x}{2}}$

Step 3.4.4

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

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{\frac{- x \cdot 2}{2} + \frac{3 x}{2}}$

Step 3.4.5

Combine the numerators over the common denominator.

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{\frac{- x \cdot 2 + 3 x}{2}}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Factor $x$ out of $- x \cdot 2 + 3 x$ .

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Step 3.4.6.1.1

Factor $x$ out of $- x \cdot 2$ .

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{\frac{x (- 1 \cdot 2) + 3 x}{2}}$

Step 3.4.6.1.2

Factor $x$ out of $3 x$ .

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{\frac{x (- 1 \cdot 2) + x \cdot 3}{2}}$

Step 3.4.6.1.3

Factor $x$ out of $x (- 1 \cdot 2) + x \cdot 3$ .

$e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{1}{2} e^{\frac{x (- 1 \cdot 2 + 3)}{2}}$

Step 3.4.6.2

Multiply $- 1$ by $2$ .

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

Step 3.4.6.3

Add $- 2$ and $3$ .

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

Step 3.4.7

Multiply $x$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Reorder factors in $e^{\frac{3 x}{2}} \frac{d y}{d x} + \frac{3 e^{\frac{3 x}{2}} y}{2} = \frac{e^{\frac{x}{2}}}{2}$ .

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

Step 4

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

$\frac{d}{d x} [e^{\frac{3 x}{2}} y] = \frac{e^{\frac{x}{2}}}{2}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{\frac{3 x}{2}} y] d x = \int \frac{e^{\frac{x}{2}}}{2} d x$

Step 6

Integrate the left side.

$e^{\frac{3 x}{2}} y = \int \frac{e^{\frac{x}{2}}}{2} d x$

Step 7

Integrate the right side.

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Step 7.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$e^{\frac{3 x}{2}} y = \frac{1}{2} \int e^{\frac{x}{2}} d x$

Step 7.2

Let $u = \frac{x}{2}$ . Then $d u = \frac{1}{2} d x$ , so $2 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.2.1

Let $u = \frac{x}{2}$ . Find $\frac{d u}{d x}$ .

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Step 7.2.1.1

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

$\frac{d}{d x} [\frac{x}{2}]$

Step 7.2.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{1}{2} \frac{d}{d x} [x]$

Step 7.2.1.3

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

$\frac{1}{2} \cdot 1$

Step 7.2.1.4

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 7.2.2

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

$e^{\frac{3 x}{2}} y = \frac{1}{2} \int e^{u} \frac{1}{\frac{1}{2}} d u$

Step 7.3

Simplify.

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Step 7.3.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$e^{\frac{3 x}{2}} y = \frac{1}{2} \int e^{u} (1 \cdot 2) d u$

Step 7.3.2

Multiply $2$ by $1$ .

$e^{\frac{3 x}{2}} y = \frac{1}{2} \int e^{u} \cdot 2 d u$

Step 7.3.3

Move $2$ to the left of $e^{u}$ .

$e^{\frac{3 x}{2}} y = \frac{1}{2} \int 2 e^{u} d u$

Step 7.4

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

$e^{\frac{3 x}{2}} y = \frac{1}{2} (2 \int e^{u} d u)$

Step 7.5

Simplify.

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Step 7.5.1

Combine $2$ and $\frac{1}{2}$ .

$e^{\frac{3 x}{2}} y = \frac{2}{2} \int e^{u} d u$

Step 7.5.2

Cancel the common factor of $2$ .

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Step 7.5.2.1

Cancel the common factor.

$e^{\frac{3 x}{2}} y = \frac{2}{2} \int e^{u} d u$

Step 7.5.2.2

Rewrite the expression.

$e^{\frac{3 x}{2}} y = 1 \int e^{u} d u$

Step 7.5.3

Multiply $\int e^{u} d u$ by $1$ .

$e^{\frac{3 x}{2}} y = \int e^{u} d u$

Step 7.6

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

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

Step 7.7

Replace all occurrences of $u$ with $\frac{x}{2}$ .

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

Step 7.8

Reorder terms.

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

Step 8

Divide each term in $e^{\frac{3 x}{2}} y = e^{\frac{1}{2} x} + C$ by $e^{\frac{3 x}{2}}$ and simplify.

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Step 8.1

Divide each term in $e^{\frac{3 x}{2}} y = e^{\frac{1}{2} x} + C$ by $e^{\frac{3 x}{2}}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $e^{\frac{3 x}{2}}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $e^{\frac{1}{2} x}$ and $e^{\frac{3 x}{2}}$ .

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Step 8.3.1.1.1

Factor $e^{\frac{3 x}{2}}$ out of $e^{\frac{1}{2} x}$ .

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

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

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

Step 8.3.1.1.2.2

Cancel the common factor.

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

Step 8.3.1.1.2.3

Rewrite the expression.

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

Step 8.3.1.1.2.4

Divide $e^{\frac{\frac{1}{2} x \cdot 2 - 3 x}{2}}$ by $1$ .

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

Step 8.3.1.2

Simplify the numerator.

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Step 8.3.1.2.1

Factor $x$ out of $\frac{1}{2} x \cdot 2 - 3 x$ .

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Step 8.3.1.2.1.1

Factor $x$ out of $\frac{1}{2} x \cdot 2$ .

$y = e^{\frac{x (\frac{1}{2} \cdot 2) - 3 x}{2}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.2.1.2

Factor $x$ out of $- 3 x$ .

$y = e^{\frac{x (\frac{1}{2} \cdot 2) + x \cdot - 3}{2}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.2.1.3

Factor $x$ out of $x (\frac{1}{2} \cdot 2) + x \cdot - 3$ .

$y = e^{\frac{x (\frac{1}{2} \cdot 2 - 3)}{2}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.2.2

Cancel the common factor of $2$ .

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Step 8.3.1.2.2.1

Cancel the common factor.

$y = e^{\frac{x (\frac{1}{2} \cdot 2 - 3)}{2}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.2.2.2

Rewrite the expression.

$y = e^{\frac{x (1 - 3)}{2}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.2.3

Subtract $3$ from $1$ .

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

Step 8.3.1.3

Cancel the common factor of $- 2$ and $2$ .

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Step 8.3.1.3.1

Factor $2$ out of $x \cdot - 2$ .

$y = e^{\frac{2 (x \cdot - 1)}{2}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.3.2

Cancel the common factors.

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Step 8.3.1.3.2.1

Factor $2$ out of $2$ .

$y = e^{\frac{2 (x \cdot - 1)}{2 (1)}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.3.2.2

Cancel the common factor.

$y = e^{\frac{2 (x \cdot - 1)}{2 \cdot 1}} + \frac{C}{e^{\frac{3 x}{2}}}$

Step 8.3.1.3.2.3

Rewrite the expression.

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

Step 8.3.1.3.2.4

Divide $x \cdot - 1$ by $1$ .

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

Step 8.3.1.4

Move $- 1$ to the left of $x$ .

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

Step 8.3.1.5

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

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

Step 9

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $5$ for $y$ in $y = e^{- x} + \frac{C}{e^{\frac{3 x}{2}}}$ .

$5 = e^{0} + \frac{C}{e^{\frac{3 \cdot 0}{2}}}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $e^{0} + \frac{C}{e^{\frac{3 \cdot 0}{2}}} = 5$ .

$e^{0} + \frac{C}{e^{\frac{3 \cdot 0}{2}}} = 5$

Step 10.2

Simplify each term.

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Step 10.2.1

Anything raised to $0$ is $1$ .

$1 + \frac{C}{e^{\frac{3 \cdot 0}{2}}} = 5$

Step 10.2.2

Cancel the common factor of $0$ and $2$ .

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Step 10.2.2.1

Factor $2$ out of $3 \cdot 0$ .

$1 + \frac{C}{e^{\frac{2 (3 \cdot 0)}{2}}} = 5$

Step 10.2.2.2

Cancel the common factors.

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Step 10.2.2.2.1

Factor $2$ out of $2$ .

$1 + \frac{C}{e^{\frac{2 (3 \cdot 0)}{2 (1)}}} = 5$

Step 10.2.2.2.2

Cancel the common factor.

$1 + \frac{C}{e^{\frac{2 (3 \cdot 0)}{2 \cdot 1}}} = 5$

Step 10.2.2.2.3

Rewrite the expression.

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

Step 10.2.2.2.4

Divide $3 \cdot 0$ by $1$ .

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

Step 10.2.3

Simplify the denominator.

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Step 10.2.3.1

Multiply $3$ by $0$ .

$1 + \frac{C}{e^{0}} = 5$

Step 10.2.3.2

Anything raised to $0$ is $1$ .

$1 + \frac{C}{1} = 5$

Step 10.2.4

Divide $C$ by $1$ .

$1 + C = 5$

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 $1$ from both sides of the equation.

$C = 5 - 1$

Step 10.3.2

Subtract $1$ from $5$ .

$C = 4$

Step 11

Substitute $4$ for $C$ in $y = e^{- x} + \frac{C}{e^{\frac{3 x}{2}}}$ and simplify.

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Step 11.1

Substitute $4$ for $C$ .

$y = e^{- x} + \frac{4}{e^{\frac{3 x}{2}}}$