Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Combine.

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

Step 1.2.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 1.2.2.1

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

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Step 1.2.2.1.1

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

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

Step 1.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.2.2

Add $1$ and $2$ .

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

Step 1.2.3

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 1.2.3.1

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

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Step 1.2.3.1.1

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

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

Step 1.2.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.3.2

Add $2$ and $1$ .

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

Step 1.2.4

Cancel the common factor of $y^{3}$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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 $1$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Cancel the common factors.

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

Cancel the common factor.

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

Step 2.2.1.3.3

Rewrite the expression.

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

Step 2.2.2

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

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

$e^{x + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

$| y | = e^{x + C}$

Step 3.3.2

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

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

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{x + C}$ as $e^{x} e^{C}$ .

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

Step 4.2

Reorder $e^{x}$ and $e^{C}$ .

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

Step 4.3

Combine constants with the plus or minus.

$y = C e^{x}$