Calculus Examples

$\frac{d y}{d x} = 4 x^{2} y - 8 x y$

Step 1

Separate the variables.

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Step 1.1

Factor $4 x y$ out of $4 x^{2} y - 8 x y$ .

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Step 1.1.1

Factor $4 x y$ out of $4 x^{2} y$ .

$\frac{d y}{d x} = 4 x y (x) - 8 x y$

Step 1.1.2

Factor $4 x y$ out of $- 8 x y$ .

$\frac{d y}{d x} = 4 x y (x) + 4 x y (- 2)$

Step 1.1.3

Factor $4 x y$ out of $4 x y (x) + 4 x y (- 2)$ .

$\frac{d y}{d x} = 4 x y (x - 2)$

Step 1.2

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

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (4 x y (x - 2))$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y} \frac{d y}{d x} = 4 \frac{1}{y} (x y (x - 2))$

Step 1.3.2

Combine $4$ and $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{4}{y} (x y (x - 2))$

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

Factor $y$ out of $x y (x - 2)$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{4}{y} (y (x (x - 2)))$

Step 1.3.3.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{4}{y} (y (x (x - 2)))$

Step 1.3.3.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = 4 (x (x - 2))$

Step 1.3.4

Apply the distributive property.

$\frac{1}{y} \frac{d y}{d x} = 4 (x \cdot x + x \cdot - 2)$

Step 1.3.5

Multiply $x$ by $x$ .

$\frac{1}{y} \frac{d y}{d x} = 4 (x^{2} + x \cdot - 2)$

Step 1.3.6

Move $- 2$ to the left of $x$ .

$\frac{1}{y} \frac{d y}{d x} = 4 (x^{2} - 2 \cdot x)$

Step 1.3.7

Apply the distributive property.

$\frac{1}{y} \frac{d y}{d x} = 4 x^{2} + 4 (- 2 x)$

Step 1.3.8

Multiply $- 2$ by $4$ .

$\frac{1}{y} \frac{d y}{d x} = 4 x^{2} - 8 x$

Step 1.4

Rewrite the equation.

$\frac{1}{y} d y = (4 x^{2} - 8 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int 4 x^{2} - 8 x d x$

Step 2.2

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

$\ln (| y |) + C_{1} = \int 4 x^{2} - 8 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\ln (| y |) + C_{1} = \int 4 x^{2} d x + \int - 8 x d x$

Step 2.3.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\ln (| y |) + C_{1} = 4 \int x^{2} d x + \int - 8 x d x$

Step 2.3.3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\ln (| y |) + C_{1} = 4 (\frac{1}{3} x^{3} + C_{2}) + \int - 8 x d x$

Step 2.3.4

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$\ln (| y |) + C_{1} = 4 (\frac{1}{3} x^{3} + C_{2}) - 8 \int x d x$

Step 2.3.5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\ln (| y |) + C_{1} = 4 (\frac{1}{3} x^{3} + C_{2}) - 8 (\frac{1}{2} x^{2} + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$\ln (| y |) + C_{1} = 4 (\frac{1}{3}) x^{3} - 8 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Combine $4$ and $\frac{1}{3}$ .

$\ln (| y |) + C_{1} = \frac{4}{3} x^{3} - 8 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.6.2.2

Combine $- 8$ and $\frac{1}{2}$ .

$\ln (| y |) + C_{1} = \frac{4}{3} x^{3} + \frac{- 8}{2} x^{2} + C_{4}$

Step 2.3.6.2.3

Cancel the common factor of $- 8$ and $2$ .

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Step 2.3.6.2.3.1

Factor $2$ out of $- 8$ .

$\ln (| y |) + C_{1} = \frac{4}{3} x^{3} + \frac{2 \cdot - 4}{2} x^{2} + C_{4}$

Step 2.3.6.2.3.2

Cancel the common factors.

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Step 2.3.6.2.3.2.1

Factor $2$ out of $2$ .

$\ln (| y |) + C_{1} = \frac{4}{3} x^{3} + \frac{2 \cdot - 4}{2 (1)} x^{2} + C_{4}$

Step 2.3.6.2.3.2.2

Cancel the common factor.

$\ln (| y |) + C_{1} = \frac{4}{3} x^{3} + \frac{2 \cdot - 4}{2 \cdot 1} x^{2} + C_{4}$

Step 2.3.6.2.3.2.3

Rewrite the expression.

$\ln (| y |) + C_{1} = \frac{4}{3} x^{3} + \frac{- 4}{1} x^{2} + C_{4}$

Step 2.3.6.2.3.2.4

Divide $- 4$ by $1$ .

$\ln (| y |) + C_{1} = \frac{4}{3} x^{3} - 4 x^{2} + C_{4}$

Step 2.3.7

Reorder terms.

$\ln (| y |) + C_{1} = - 4 x^{2} + \frac{4}{3} x^{3} + C_{4}$

Step 2.4

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

$\ln (| y |) = - 4 x^{2} + \frac{4}{3} x^{3} + C$

Step 3

Solve for $y$ .

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Step 3.1

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

$e^{\ln (| y |)} = e^{- 4 x^{2} + \frac{4}{3} x^{3} + C}$

Step 3.2

Rewrite $\ln (| y |) = - 4 x^{2} + \frac{4}{3} x^{3} + C$ 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 $b^{y} = x$ .

$e^{- 4 x^{2} + \frac{4}{3} x^{3} + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{- 4 x^{2} + \frac{4}{3} x^{3} + C}$ .

$| y | = e^{- 4 x^{2} + \frac{4}{3} x^{3} + C}$

Step 3.3.2

Combine $\frac{4}{3}$ and $x^{3}$ .

$| y | = e^{- 4 x^{2} + \frac{4 x^{3}}{3} + C}$

Step 3.3.3

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

$y = \pm e^{- 4 x^{2} + \frac{4 x^{3}}{3} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{- 4 x^{2} + \frac{4 x^{3}}{3} + C}$ as $e^{- 4 x^{2} + \frac{4 x^{3}}{3}} e^{C}$ .

$y = \pm e^{- 4 x^{2} + \frac{4 x^{3}}{3}} e^{C}$

Step 4.2

Reorder $e^{- 4 x^{2} + \frac{4 x^{3}}{3}}$ and $e^{C}$ .

$y = \pm e^{C} e^{- 4 x^{2} + \frac{4 x^{3}}{3}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{- 4 x^{2} + \frac{4 x^{3}}{3}}$