Calculus Examples

$y' \ln (| y |) = e^{x}$

Step 1

Rewrite the differential equation.

$\frac{d y}{d x} \ln (| y |) = e^{x}$

Step 2

Separate the variables.

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Step 2.1

Divide each term in $\frac{d y}{d x} \ln (| y |) = e^{x}$ by $\ln (| y |)$ and simplify.

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Step 2.1.1

Divide each term in $\frac{d y}{d x} \ln (| y |) = e^{x}$ by $\ln (| y |)$ .

$\frac{\frac{d y}{d x} \ln (| y |)}{\ln (| y |)} = \frac{e^{x}}{\ln (| y |)}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $\ln (| y |)$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} \ln (| y |)}{\ln (| y |)} = \frac{e^{x}}{\ln (| y |)}$

Step 2.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{e^{x}}{\ln (| y |)}$

Step 2.2

Multiply both sides by $\ln (| y |)$ .

$\ln (| y |) \frac{d y}{d x} = \ln (| y |) \frac{e^{x}}{\ln (| y |)}$

Step 2.3

Cancel the common factor of $\ln (| y |)$ .

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Step 2.3.1

Cancel the common factor.

$\ln (| y |) \frac{d y}{d x} = \ln (| y |) \frac{e^{x}}{\ln (| y |)}$

Step 2.3.2

Rewrite the expression.

$\ln (| y |) \frac{d y}{d x} = e^{x}$

Step 2.4

Rewrite the equation.

$\ln (| y |) d y = e^{x} d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int \ln (| y |) d y = \int e^{x} d x$

Step 3.2

Integrate the left side.

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Step 3.2.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (| y |)$ and $d v = 1$ .

$\ln (| y |) y - \int y \frac{1}{y} d y = \int e^{x} d x$

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

$\ln (| y |) y - \int \frac{y}{y} d y = \int e^{x} d x$

Step 3.2.2.2

Cancel the common factor of $y$ .

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Step 3.2.2.2.1

Cancel the common factor.

$\ln (| y |) y - \int \frac{y}{y} d y = \int e^{x} d x$

Step 3.2.2.2.2

Rewrite the expression.

$\ln (| y |) y - \int d y = \int e^{x} d x$

Step 3.2.3

Apply the constant rule.

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

Step 3.2.4

Simplify.

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

Step 3.2.5

Reorder terms.

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

Step 3.3

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

$y \ln (| y |) - y + C_{1} = e^{x} + C_{2}$

Step 3.4

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

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