Calculus Examples

$\frac{d y}{d x} = \frac{x^{2} y^{2}}{x y}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{x^{2}}{x} \cdot \frac{y^{2}}{y}$

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $y$ and $y^{2}$ .

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Step 1.3.1.1

Raise $y$ to the power of $1$ .

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

Step 1.3.1.2

Factor $y$ out of $y^{1}$ .

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

Step 1.3.1.3

Cancel the common factors.

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Step 1.3.1.3.1

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

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

Step 1.3.1.3.2

Cancel the common factor.

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

Step 1.3.1.3.3

Rewrite the expression.

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

Step 1.3.2

Combine.

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

Step 1.3.3

Combine.

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

Step 1.3.4

Multiply $y$ by $y$ by adding the exponents.

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Step 1.3.4.1

Move $y$ .

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

Step 1.3.4.2

Multiply $y$ by $y$ .

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

Step 1.3.5

Multiply $x^{2} y^{2}$ by $1$ .

$\frac{y}{y^{2}} \frac{d y}{d x} = \frac{x^{2} y^{2}}{y^{2} x}$

Step 1.3.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.3.6.1

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

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

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

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

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

Step 1.3.6.2.2

Cancel the common factor.

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

Step 1.3.6.2.3

Rewrite the expression.

$\frac{y}{y^{2}} \frac{d y}{d x} = \frac{x y^{2}}{y^{2}}$

Step 1.3.7

Cancel the common factor of $y^{2}$ .

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Step 1.3.7.1

Cancel the common factor.

$\frac{y}{y^{2}} \frac{d y}{d x} = \frac{x y^{2}}{y^{2}}$

Step 1.3.7.2

Divide $x$ by $1$ .

$\frac{y}{y^{2}} \frac{d y}{d x} = x$

Step 1.4

Rewrite the equation.

$\frac{y}{y^{2}} d y = x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{y}{y^{2}} d y = \int x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Cancel the common factor of $y$ and $y^{2}$ .

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Step 2.2.1.1

Raise $y$ to the power of $1$ .

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

Step 2.2.1.2

Factor $y$ out of $y^{1}$ .

$\int \frac{y \cdot 1}{y^{2}} d y = \int x d x$

Step 2.2.1.3

Cancel the common factors.

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

Cancel the common factor.

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

Step 2.2.1.3.3

Rewrite the expression.

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Rewrite $\ln (| y |) = \frac{1}{2} x^{2} + C$ 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 $b^{y} = x$ .

$e^{\frac{1}{2} x^{2} + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{\frac{1}{2} x^{2} + C}$ .

$| y | = e^{\frac{1}{2} x^{2} + C}$

Step 3.3.2

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 3.3.3

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

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

Step 4

Group the constant terms together.

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Step 4.1

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

$y = \pm e^{\frac{x^{2}}{2}} e^{C}$

Step 4.2

Reorder $e^{\frac{x^{2}}{2}}$ and $e^{C}$ .

$y = \pm e^{C} e^{\frac{x^{2}}{2}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{\frac{x^{2}}{2}}$