Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Add $\frac{y}{x}$ to both sides of the equation.

$\frac{d y}{d x} = y x^{4} + \frac{y}{x}$

Step 1.2

Factor.

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Step 1.2.1

Factor $y$ out of $y x^{4} + \frac{y}{x}$ .

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Step 1.2.1.1

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

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

Step 1.2.1.2

Factor $y$ out of $\frac{y}{x}$ .

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

Step 1.2.1.3

Factor $y$ out of $y (x^{4}) + y \frac{1}{x}$ .

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

Step 1.2.2

To write $x^{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

$\frac{d y}{d x} = y \frac{x^{4} x + 1}{x}$

Step 1.2.4

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.2.4.1

Multiply $x^{4}$ by $x$ .

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Step 1.2.4.1.1

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

$\frac{d y}{d x} = y \frac{x^{4} x^{1} + 1}{x}$

Step 1.2.4.1.2

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

$\frac{d y}{d x} = y \frac{x^{4 + 1} + 1}{x}$

Step 1.2.4.2

Add $4$ and $1$ .

$\frac{d y}{d x} = y \frac{x^{5} + 1}{x}$

Step 1.3

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

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \frac{x^{5} + 1}{x})$

Step 1.4

Cancel the common factor of $y$ .

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Step 1.4.1

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \frac{x^{5} + 1}{x})$

Step 1.4.2

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = \frac{x^{5} + 1}{x}$

Step 1.5

Rewrite the equation.

$\frac{1}{y} d y = \frac{x^{5} + 1}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int \frac{x^{5} + 1}{x} d x$

Step 2.2

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

$\ln (| y |) + C_{1} = \int \frac{x^{5} + 1}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

$\ln (| y |) + C_{1} = \int \frac{x^{5}}{x} + \frac{1}{x} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$\ln (| y |) + C_{1} = \int \frac{x^{5}}{x} d x + \int \frac{1}{x} d x$

Step 2.3.3

Cancel the common factor of $x^{5}$ and $x$ .

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Step 2.3.3.1

Factor $x$ out of $x^{5}$ .

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

Step 2.3.3.2

Cancel the common factors.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

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

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

Step 2.3.3.2.3

Cancel the common factor.

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

Step 2.3.3.2.4

Rewrite the expression.

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

Step 2.3.3.2.5

Divide $x^{4}$ by $1$ .

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

Step 2.3.4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\ln (| y |) + C_{1} = \frac{1}{5} x^{5} + C_{2} + \int \frac{1}{x} d x$

Step 2.3.5

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

$\ln (| y |) + C_{1} = \frac{1}{5} x^{5} + C_{2} + \ln (| x |) + C_{3}$

Step 2.3.6

Simplify.

$\ln (| y |) + C_{1} = \frac{1}{5} x^{5} + \ln (| x |) + C_{4}$

Step 2.4

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

$\ln (| y |) = \frac{1}{5} x^{5} + \ln (| x |) + C$

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 $\frac{1}{5}$ and $x^{5}$ .

$\ln (| y |) = \frac{x^{5}}{5} + \ln (| x |) + C$

Step 3.2

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

$\ln (| y |) - \ln (| x |) = \frac{x^{5}}{5} + C$

Step 3.3

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

$\ln (\frac{| y |}{| x |}) = \frac{x^{5}}{5} + C$

Step 3.4

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

$e^{\ln (\frac{| y |}{| x |})} = e^{\frac{x^{5}}{5} + C}$

Step 3.5

Rewrite $\ln (\frac{| y |}{| x |}) = \frac{x^{5}}{5} + C$ 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 $b^{y} = x$ .

$e^{\frac{x^{5}}{5} + C} = \frac{| y |}{| x |}$

Step 3.6

Solve for $y$ .

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Step 3.6.1

Rewrite the equation as $\frac{| y |}{| x |} = e^{\frac{x^{5}}{5} + C}$ .

$\frac{| y |}{| x |} = e^{\frac{x^{5}}{5} + C}$

Step 3.6.2

Multiply both sides by $| x |$ .

$\frac{| y |}{| x |} | x | = e^{\frac{x^{5}}{5} + C} | x |$

Step 3.6.3

Simplify the left side.

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Step 3.6.3.1

Cancel the common factor of $| x |$ .

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Step 3.6.3.1.1

Cancel the common factor.

$\frac{| y |}{| x |} | x | = e^{\frac{x^{5}}{5} + C} | x |$

Step 3.6.3.1.2

Rewrite the expression.

$| y | = e^{\frac{x^{5}}{5} + C} | x |$

Step 3.6.4

Solve for $y$ .

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Step 3.6.4.1

Reorder factors in $e^{\frac{x^{5}}{5} + C} | x |$ .

$| y | = | x | e^{\frac{x^{5}}{5} + C}$

Step 3.6.4.2

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

$y = \pm | x | e^{\frac{x^{5}}{5} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{\frac{x^{5}}{5} + C}$ as $e^{\frac{x^{5}}{5}} e^{C}$ .

$y = \pm | x | (e^{\frac{x^{5}}{5}} e^{C})$

Step 4.2

Reorder $e^{\frac{x^{5}}{5}}$ and $e^{C}$ .

$y = \pm | x | (e^{C} e^{\frac{x^{5}}{5}})$

Step 4.3

Combine constants with the plus or minus.

$y = C | x | e^{\frac{x^{5}}{5}}$