Calculus Examples

$y' = 3 x^{2}$ , $y = x^{3} - 4 + c$ , $y (0) = 5$

Step 1

Verify that the given solution satisfies the differential equation.

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Step 1.1

Find $y'$ .

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Step 1.1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{3} - 4 + c)$

Step 1.1.2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 1.1.3

Differentiate the right side of the equation.

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Step 1.1.3.1

By the Sum Rule, the derivative of $x^{3} - 4 + c$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4] + \frac{d}{d x} [c]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4] + \frac{d}{d x} [c]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 4] + \frac{d}{d x} [c]$

Step 1.1.3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$3 x^{2} + 0 + \frac{d}{d x} [c]$

Step 1.1.3.4

Since $c$ is constant with respect to $x$ , the derivative of $c$ with respect to $x$ is $0$ .

$3 x^{2} + 0 + 0$

Step 1.1.3.5

Combine terms.

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Step 1.1.3.5.1

Add $3 x^{2}$ and $0$ .

$3 x^{2} + 0$

Step 1.1.3.5.2

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.1.4

Reform the equation by setting the left side equal to the right side.

$y' = 3 x^{2}$

Step 1.2

Substitute into the given differential equation.

$3 x^{2} = 3 x^{2}$

Step 1.3

The given solution satisfies the given differential equation.

$y = x^{3} - 4 + c$ is a solution to $y' = 3 x^{2}$

Step 2

Substitute in the initial condition.

$5 = 0^{3} - 4 + c$

Step 3

Solve for $c$ .

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Step 3.1

Rewrite the equation as $0^{3} - 4 + c = 5$ .

$0^{3} - 4 + c = 5$

Step 3.2

Simplify $0^{3} - 4 + c$ .

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Step 3.2.1

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

$0 - 4 + c = 5$

Step 3.2.2

Subtract $4$ from $0$ .

$- 4 + c = 5$

Step 3.3

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

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Step 3.3.1

Add $4$ to both sides of the equation.

$c = 5 + 4$

Step 3.3.2

Add $5$ and $4$ .

$c = 9$