Calculus Examples

$y' = y x^{3}$

Step 1

Rewrite the differential equation.

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

Step 2

Separate the variables.

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Step 2.1

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

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

Step 2.2

Cancel the common factor of $y$ .

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Step 2.2.1

Factor $y$ out of $y x^{3}$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

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

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

Step 3.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\ln (| y |) + C_{1} = \frac{1}{4} x^{4} + C_{2}$

Step 3.4

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

$\ln (| y |) = \frac{1}{4} x^{4} + C$

Step 4

Solve for $y$ .

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Step 4.1

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

$e^{\ln (| y |)} = e^{\frac{1}{4} x^{4} + C}$

Step 4.2

Rewrite $\ln (| y |) = \frac{1}{4} x^{4} + C$ 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 $b^{y} = x$ .

$e^{\frac{1}{4} x^{4} + C} = | y |$

Step 4.3

Solve for $y$ .

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Step 4.3.1

Rewrite the equation as $| y | = e^{\frac{1}{4} x^{4} + C}$ .

$| y | = e^{\frac{1}{4} x^{4} + C}$

Step 4.3.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$| y | = e^{\frac{x^{4}}{4} + C}$

Step 4.3.3

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

$y = \pm e^{\frac{x^{4}}{4} + C}$

Step 5

Group the constant terms together.

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Step 5.1

Rewrite $e^{\frac{x^{4}}{4} + C}$ as $e^{\frac{x^{4}}{4}} e^{C}$ .

$y = \pm e^{\frac{x^{4}}{4}} e^{C}$

Step 5.2

Reorder $e^{\frac{x^{4}}{4}}$ and $e^{C}$ .

$y = \pm e^{C} e^{\frac{x^{4}}{4}}$

Step 5.3

Combine constants with the plus or minus.

$y = C e^{\frac{x^{4}}{4}}$