Calculus Examples

Solve the equation $\frac{d y}{d x} = 2 y \cos (x)$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (2 y \cos (x))$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y} \frac{d y}{d x} = 2 \frac{1}{y} (y \cos (x))$

Step 1.2.2

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

$\frac{1}{y} \frac{d y}{d x} = \frac{2}{y} (y \cos (x))$

Step 1.2.3

Cancel the common factor of $y$ .

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Step 1.2.3.1

Factor $y$ out of $y \cos (x)$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{2}{y} (y (\cos (x)))$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{2}{y} (y \cos (x))$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = 2 \cos (x)$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = 2 \cos (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int 2 \cos (x) d x$

Step 2.2

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$\ln (| y |) + C_{1} = \int 2 \cos (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\ln (| y |) + C_{1} = 2 \int \cos (x) d x$

Step 2.3.2

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$\ln (| y |) + C_{1} = 2 (\sin (x) + C_{2})$

Step 2.3.3

Simplify.

$\ln (| y |) + C_{1} = 2 \sin (x) + C_{2}$

Step 2.4

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

$\ln (| y |) = 2 \sin (x) + C$

Step 3

Solve for $y$ .

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Step 3.1

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y |)} = e^{2 \sin (x) + C}$

Step 3.2

Rewrite $\ln (| y |) = 2 \sin (x) + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{2 \sin (x) + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{2 \sin (x) + C}$ .

$| y | = e^{2 \sin (x) + C}$

Step 3.3.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm e^{2 \sin (x) + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{2 \sin (x) + C}$ as $e^{2 \sin (x)} e^{C}$ .

$y = \pm e^{2 \sin (x)} e^{C}$

Step 4.2

Reorder $e^{2 \sin (x)}$ and $e^{C}$ .

$y = \pm e^{C} e^{2 \sin (x)}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{2 \sin (x)}$