Calculus Examples Differential Equations

Solve the Differential Equation

$\frac{d y}{d x} = e^{- 11 x}$ , $(0, 13)$

Step 1

Rewrite the equation.

$d y = e^{- 11 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int e^{- 11 x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int e^{- 11 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = - 11 x$ . Then $d u = - 11 d x$ , so $- \frac{1}{11} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = - 11 x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $- 11 x$ .

$\frac{d}{d x} [- 11 x]$

Step 2.3.1.1.2

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$- 11 \frac{d}{d x} [x]$

Step 2.3.1.1.3

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

$- 11 \cdot 1$

Step 2.3.1.1.4

Multiply $- 11$ by $1$ .

$- 11$

Step 2.3.1.2

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

$y + C_{1} = \int e^{u} \frac{1}{- 11} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move the negative in front of the fraction.

$y + C_{1} = \int e^{u} (- \frac{1}{11}) d u$

Step 2.3.2.2

Combine $e^{u}$ and $\frac{1}{11}$ .

$y + C_{1} = \int - \frac{e^{u}}{11} d u$

Step 2.3.3

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

$y + C_{1} = - \int \frac{e^{u}}{11} d u$

Step 2.3.4

Since $\frac{1}{11}$ is constant with respect to $u$ , move $\frac{1}{11}$ out of the integral.

$y + C_{1} = - (\frac{1}{11} \int e^{u} d u)$

Step 2.3.5

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

$y + C_{1} = - \frac{1}{11} (e^{u} + C_{2})$

Step 2.3.6

Simplify.

$y + C_{1} = - \frac{1}{11} e^{u} + C_{2}$

Step 2.3.7

Replace all occurrences of $u$ with $- 11 x$ .

$y + C_{1} = - \frac{1}{11} e^{- 11 x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = - \frac{1}{11} e^{- 11 x} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $13$ for $y$ in $y = - \frac{1}{11} e^{- 11 x} + K$ .

$13 = - \frac{1}{11} e^{- 11 \cdot 0} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{1}{11} \cdot e^{- 11 \cdot 0} + K = 13$ .

$- \frac{1}{11} \cdot e^{- 11 \cdot 0} + K = 13$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $- 11$ by $0$ .

$- \frac{1}{11} \cdot e^{0} + K = 13$

Step 4.2.2

Anything raised to $0$ is $1$ .

$- \frac{1}{11} \cdot 1 + K = 13$

Step 4.2.3

Multiply $- 1$ by $1$ .

$- \frac{1}{11} + K = 13$

Step 4.3

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

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Step 4.3.1

Add $\frac{1}{11}$ to both sides of the equation.

$K = 13 + \frac{1}{11}$

Step 4.3.2

To write $13$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$K = 13 \cdot \frac{11}{11} + \frac{1}{11}$

Step 4.3.3

Combine $13$ and $\frac{11}{11}$ .

$K = \frac{13 \cdot 11}{11} + \frac{1}{11}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{13 \cdot 11 + 1}{11}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $13$ by $11$ .

$K = \frac{143 + 1}{11}$

Step 4.3.5.2

Add $143$ and $1$ .

$K = \frac{144}{11}$

Step 5

Substitute $\frac{144}{11}$ for $K$ in $y = - \frac{1}{11} e^{- 11 x} + K$ and simplify.

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Step 5.1

Substitute $\frac{144}{11}$ for $K$ .

$y = - \frac{1}{11} e^{- 11 x} + \frac{144}{11}$

Step 5.2

Combine $e^{- 11 x}$ and $\frac{1}{11}$ .

$y = - \frac{e^{- 11 x}}{11} + \frac{144}{11}$

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