Calculus Examples

$\frac{d^{2} y}{d x^{2}} + 4 y = \cos (2 x)$

Step 1

Assume all solutions are of the form $y = e^{r x}$ .

Step 2

Find the characteristic equation for $\frac{d^{2} y}{d x^{2}} + 4 y = \cos (2 x)$ .

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Step 2.1

Find the first derivative.

$\frac{d y}{d x} = r e^{r x}$

Step 2.2

Find the second derivative.

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

Step 2.3

Substitute into the differential equation.

$r^{2} e^{r x} + 4 e^{r x} = \cos (2 x)$

Step 2.4

Factor out $e^{r x}$ .

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Step 2.4.1

Factor $e^{r x}$ out of $r^{2} e^{r x}$ .

$e^{r x} r^{2} + 4 e^{r x} = \cos (2 x)$

Step 2.4.2

Factor $e^{r x}$ out of $4 e^{r x}$ .

$e^{r x} r^{2} + e^{r x} \cdot 4 = \cos (2 x)$

Step 2.4.3

Factor $e^{r x}$ out of $e^{r x} r^{2} + e^{r x} \cdot 4$ .

$e^{r x} (r^{2} + 4) = \cos (2 x)$

Step 2.5

Since exponentials can never be zero, divide both sides by $e^{r x}$ .

$r^{2} + 4 = \cos (2 x)$

Step 3

Solve for $r$ .

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Step 3.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$r^{2} + 4 = 1 - 2 \sin^{2} (x)$

Step 3.2

Subtract $4$ from both sides of the equation.

$r^{2} = 1 - 2 \sin^{2} (x) - 4$

Step 3.3

Simplify the right side.

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Step 3.3.1

Subtract $4$ from $1$ .

$r^{2} = - 3 - 2 \sin^{2} (x)$

Step 3.4

Solve the equation for $r$ .

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Step 3.4.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{- 3 - 2 \sin^{2} (x)}$

Step 3.4.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.2.1

First, use the positive value of the $\pm$ to find the first solution.

$r = \sqrt{- 3 - 2 \sin^{2} (x)}$

Step 3.4.2.2

Next, use the negative value of the $\pm$ to find the second solution.