Calculus Examples

$x \frac{d y}{d x} = \frac{y}{2 + 3 y}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x \frac{d y}{d x} = \frac{y}{2 + 3 y}$ by $x$ and simplify.

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Step 1.1.1

Divide each term in $x \frac{d y}{d x} = \frac{y}{2 + 3 y}$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} = \frac{\frac{y}{2 + 3 y}}{x}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} = \frac{\frac{y}{2 + 3 y}}{x}$

Step 1.1.2.1.2

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

$\frac{d y}{d x} = \frac{\frac{y}{2 + 3 y}}{x}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d y}{d x} = \frac{y}{2 + 3 y} \cdot \frac{1}{x}$

Step 1.1.3.2

Multiply $\frac{y}{2 + 3 y}$ by $\frac{1}{x}$ .

$\frac{d y}{d x} = \frac{y}{(2 + 3 y) x}$

Step 1.1.3.3

Reorder factors in $\frac{y}{(2 + 3 y) x}$ .

$\frac{d y}{d x} = \frac{y}{x (2 + 3 y)}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{y}{2 + 3 y} \cdot \frac{1}{x}$

Step 1.3

Multiply both sides by $\frac{2 + 3 y}{y}$ .

$\frac{2 + 3 y}{y} \frac{d y}{d x} = \frac{2 + 3 y}{y} (\frac{y}{2 + 3 y} \cdot \frac{1}{x})$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{y}{2 + 3 y}$ by $\frac{1}{x}$ .

$\frac{2 + 3 y}{y} \frac{d y}{d x} = \frac{2 + 3 y}{y} \cdot \frac{y}{(2 + 3 y) x}$

Step 1.4.2

Cancel the common factor of $2 + 3 y$ .

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Step 1.4.2.1

Factor $2 + 3 y$ out of $(2 + 3 y) x$ .

$\frac{2 + 3 y}{y} \frac{d y}{d x} = \frac{2 + 3 y}{y} \cdot \frac{y}{(2 + 3 y) (x)}$

Step 1.4.2.2

Cancel the common factor.

$\frac{2 + 3 y}{y} \frac{d y}{d x} = \frac{2 + 3 y}{y} \cdot \frac{y}{(2 + 3 y) x}$

Step 1.4.2.3

Rewrite the expression.

$\frac{2 + 3 y}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y}{x}$

Step 1.4.3

Cancel the common factor of $y$ .

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Step 1.4.3.1

Cancel the common factor.

$\frac{2 + 3 y}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y}{x}$

Step 1.4.3.2

Rewrite the expression.

$\frac{2 + 3 y}{y} \frac{d y}{d x} = \frac{1}{x}$

Step 1.5

Rewrite the equation.

$\frac{2 + 3 y}{y} d y = \frac{1}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{2 + 3 y}{y} d y = \int \frac{1}{x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the fraction into multiple fractions.

$\int \frac{2}{y} + \frac{3 y}{y} d y = \int \frac{1}{x} d x$

Step 2.2.2

Split the single integral into multiple integrals.

$\int \frac{2}{y} d y + \int \frac{3 y}{y} d y = \int \frac{1}{x} d x$

Step 2.2.3

Cancel the common factor of $y$ .

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Step 2.2.3.1

Cancel the common factor.

$\int \frac{2}{y} d y + \int \frac{3 y}{y} d y = \int \frac{1}{x} d x$

Step 2.2.3.2

Divide $3$ by $1$ .

$\int \frac{2}{y} d y + \int 3 d y = \int \frac{1}{x} d x$

Step 2.2.4

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

$2 \int \frac{1}{y} d y + \int 3 d y = \int \frac{1}{x} d x$

Step 2.2.5

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

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

Step 2.2.6

Apply the constant rule.

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

Step 2.2.7

Simplify.

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

Step 2.2.8

Reorder terms.

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

Step 2.3

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

$3 y + 2 \ln (| y |) + C_{3} = \ln (| x |) + C_{4}$

Step 2.4

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

$3 y + 2 \ln (| y |) = \ln (| x |) + C$