Calculus Examples

$\frac{d y}{d x} = 25 e^{x} - x^{2}$ , $y (0) = 19$

Step 1

Rewrite the equation.

$d y = (25 e^{x} - x^{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 25 e^{x} - x^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 25 e^{x} - x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 25 e^{x} d x + \int - x^{2} d x$

Step 2.3.2

Since $25$ is constant with respect to $x$ , move $25$ out of the integral.

$y + C_{1} = 25 \int e^{x} d x + \int - x^{2} d x$

Step 2.3.3

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

$y + C_{1} = 25 (e^{x} + C_{2}) + \int - x^{2} d x$

Step 2.3.4

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

$y + C_{1} = 25 (e^{x} + C_{2}) - \int x^{2} d x$

Step 2.3.5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$y + C_{1} = 25 (e^{x} + C_{2}) - (\frac{1}{3} x^{3} + C_{3})$

Step 2.3.6

Simplify.

$y + C_{1} = 25 e^{x} - \frac{1}{3} x^{3} + C_{4}$

Step 2.3.7

Reorder terms.

$y + C_{1} = - \frac{1}{3} x^{3} + 25 e^{x} + C_{4}$

Step 2.4

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

$y = - \frac{1}{3} x^{3} + 25 e^{x} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $19$ for $y$ in $y = - \frac{1}{3} x^{3} + 25 e^{x} + K$ .

$19 = - \frac{1}{3} \cdot 0^{3} + 25 e^{0} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{1}{3} \cdot 0^{3} + 25 e^{0} + K = 19$ .

$- \frac{1}{3} \cdot 0^{3} + 25 e^{0} + K = 19$

Step 4.2

Simplify $- \frac{1}{3} \cdot 0^{3} + 25 e^{0} + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{3} \cdot 0 + 25 e^{0} + K = 19$

Step 4.2.1.2

Multiply $- \frac{1}{3} \cdot 0$ .

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Step 4.2.1.2.1

Multiply $0$ by $- 1$ .

$0 (\frac{1}{3}) + 25 e^{0} + K = 19$

Step 4.2.1.2.2

Multiply $0$ by $\frac{1}{3}$ .

$0 + 25 e^{0} + K = 19$

Step 4.2.1.3

Anything raised to $0$ is $1$ .

$0 + 25 \cdot 1 + K = 19$

Step 4.2.1.4

Multiply $25$ by $1$ .

$0 + 25 + K = 19$

Step 4.2.2

Add $0$ and $25$ .

$25 + K = 19$

Step 4.3

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

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Step 4.3.1

Subtract $25$ from both sides of the equation.

$K = 19 - 25$

Step 4.3.2

Subtract $25$ from $19$ .

$K = - 6$

Step 5

Substitute $- 6$ for $K$ in $y = - \frac{1}{3} x^{3} + 25 e^{x} + K$ and simplify.

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Step 5.1

Substitute $- 6$ for $K$ .

$y = - \frac{1}{3} x^{3} + 25 e^{x} - 6$

Step 5.2

Combine $x^{3}$ and $\frac{1}{3}$ .

$y = - \frac{x^{3}}{3} + 25 e^{x} - 6$