Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify.

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

Step 2.4

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

$\ln (| y |) = \frac{1}{3} \ln (| x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Simplify the right side.

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Step 3.1.1

Combine $\frac{1}{3}$ and $\ln (| x |)$ .

$\ln (| y |) = \frac{\ln (| x |)}{3} + C$

Step 3.2

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| y |) - \frac{\ln (| x |)}{3} = C$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify $\ln (| y |) - \frac{\ln (| x |)}{3}$ .

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Step 3.3.1.1

Simplify each term.

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Step 3.3.1.1.1

Rewrite $\frac{\ln (| x |)}{3}$ as $\frac{1}{3} \ln (| x |)$ .

$\ln (| y |) - (\frac{1}{3} \ln (| x |)) = C$

Step 3.3.1.1.2

Simplify $\frac{1}{3} \ln (| x |)$ by moving $\frac{1}{3}$ inside the logarithm.

$\ln (| y |) - \ln ({| x |}^{\frac{1}{3}}) = C$

Step 3.3.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| y |}{{| x |}^{\frac{1}{3}}}) = C$

Step 3.4

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

$e^{\ln (\frac{| y |}{{| x |}^{\frac{1}{3}}})} = e^{C}$

Step 3.5

Rewrite $\ln (\frac{| y |}{{| x |}^{\frac{1}{3}}}) = C$ 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 $b^{y} = x$ .

$e^{C} = \frac{| y |}{{| x |}^{\frac{1}{3}}}$

Step 3.6

Solve for $y$ .

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Step 3.6.1

Rewrite the equation as $\frac{| y |}{{| x |}^{\frac{1}{3}}} = e^{C}$ .

$\frac{| y |}{{| x |}^{\frac{1}{3}}} = e^{C}$

Step 3.6.2

Multiply both sides by ${| x |}^{\frac{1}{3}}$ .

$\frac{| y |}{{| x |}^{\frac{1}{3}}} {| x |}^{\frac{1}{3}} = e^{C} {| x |}^{\frac{1}{3}}$

Step 3.6.3

Simplify the left side.

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Step 3.6.3.1

Cancel the common factor of ${| x |}^{\frac{1}{3}}$ .

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Step 3.6.3.1.1

Cancel the common factor.

$\frac{| y |}{{| x |}^{\frac{1}{3}}} {| x |}^{\frac{1}{3}} = e^{C} {| x |}^{\frac{1}{3}}$

Step 3.6.3.1.2

Rewrite the expression.

$| y | = e^{C} {| x |}^{\frac{1}{3}}$

Step 3.6.4

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

$y = \pm e^{C} {| x |}^{\frac{1}{3}}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \pm C {| x |}^{\frac{1}{3}}$

Step 4.2

Combine constants with the plus or minus.

$y = C {| x |}^{\frac{1}{3}}$