Calculus Examples

$\frac{d y}{d x} = \frac{y}{x (y - 3)}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{y}{y - 3} \cdot \frac{1}{x}$

Step 1.2

Multiply both sides by $\frac{y - 3}{y}$ .

$\frac{y - 3}{y} \frac{d y}{d x} = \frac{y - 3}{y} (\frac{y}{y - 3} \cdot \frac{1}{x})$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $\frac{y}{y - 3}$ by $\frac{1}{x}$ .

$\frac{y - 3}{y} \frac{d y}{d x} = \frac{y - 3}{y} \cdot \frac{y}{(y - 3) x}$

Step 1.3.2

Cancel the common factor of $y - 3$ .

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Step 1.3.2.1

Factor $y - 3$ out of $(y - 3) x$ .

$\frac{y - 3}{y} \frac{d y}{d x} = \frac{y - 3}{y} \cdot \frac{y}{(y - 3) (x)}$

Step 1.3.2.2

Cancel the common factor.

$\frac{y - 3}{y} \frac{d y}{d x} = \frac{y - 3}{y} \cdot \frac{y}{(y - 3) x}$

Step 1.3.2.3

Rewrite the expression.

$\frac{y - 3}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y}{x}$

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

Cancel the common factor.

$\frac{y - 3}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y}{x}$

Step 1.3.3.2

Rewrite the expression.

$\frac{y - 3}{y} \frac{d y}{d x} = \frac{1}{x}$

Step 1.4

Rewrite the equation.

$\frac{y - 3}{y} d y = \frac{1}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{y - 3}{y} d y = \int \frac{1}{x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the fraction into multiple fractions.

$\int \frac{y}{y} + \frac{- 3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.2

Split the single integral into multiple integrals.

$\int \frac{y}{y} d y + \int \frac{- 3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Cancel the common factor of $y$ .

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Step 2.2.3.1.1

Cancel the common factor.

$\int \frac{y}{y} d y + \int \frac{- 3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.3.1.2

Rewrite the expression.

$\int d y + \int \frac{- 3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.3.2

Move the negative in front of the fraction.

$\int d y + \int - \frac{3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.4

Apply the constant rule.

$y + C_{1} + \int - \frac{3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.5

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

$y + C_{1} - \int \frac{3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.6

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$y + C_{1} - (3 \int \frac{1}{y} d y) = \int \frac{1}{x} d x$

Step 2.2.7

Multiply $3$ by $- 1$ .

$y + C_{1} - 3 \int \frac{1}{y} d y = \int \frac{1}{x} d x$

Step 2.2.8

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

$y + C_{1} - 3 (\ln (| y |) + C_{2}) = \int \frac{1}{x} d x$

Step 2.2.9

Simplify.

$y - 3 \ln (| y |) + C_{3} = \int \frac{1}{x} d x$

Step 2.3

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

$y - 3 \ln (| y |) + C_{3} = \ln (| x |) + C_{4}$

Step 2.4

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

$y - 3 \ln (| y |) = \ln (| x |) + C$