Calculus Examples

$x \cdot \frac{d y}{d x} + 3 y = \frac{1}{5} x^{2}$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Divide each term in $x \cdot \frac{d y}{d x} + 3 y = \frac{1}{5} x^{2}$ by $x$ .

$\frac{x \cdot \frac{d y}{d x}}{x} + \frac{3 y}{x} = \frac{\frac{1}{5} x^{2}}{x}$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\frac{x \cdot \frac{d y}{d x}}{x} + \frac{3 y}{x} = \frac{\frac{1}{5} x^{2}}{x}$

Step 1.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{\frac{1}{5} x^{2}}{x}$

Step 1.3

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

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Step 1.3.1

Factor $x$ out of $\frac{1}{5} x^{2}$ .

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (\frac{1}{5} x)}{x}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (\frac{1}{5} x)}{x^{1}}$

Step 1.3.2.2

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

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (\frac{1}{5} x)}{x \cdot 1}$

Step 1.3.2.3

Cancel the common factor.

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{x (\frac{1}{5} x)}{x \cdot 1}$

Step 1.3.2.4

Rewrite the expression.

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{\frac{1}{5} x}{1}$

Step 1.3.2.5

Divide $\frac{1}{5} x$ by $1$ .

$\frac{d y}{d x} + \frac{3 y}{x} = \frac{1}{5} x$

Step 1.4

Factor $y$ out of $\frac{3 y}{x}$ .

$\frac{d y}{d x} + y \frac{3}{x} = \frac{1}{5} x$

Step 1.5

Reorder $y$ and $\frac{3}{x}$ .

$\frac{d y}{d x} + \frac{3}{x} y = \frac{1}{5} x$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{3}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{3}{x} d x}$

Step 2.2

Integrate $\frac{3}{x}$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{3 \int \frac{1}{x} d x}$

Step 2.2.2

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

$e^{3 (\ln (| x |) + C)}$

Step 2.2.3

Simplify.

$e^{3 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{3 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{3})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{3}$

Step 3

Multiply each term by the integrating factor $x^{3}$ .

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Step 3.1

Multiply each term by $x^{3}$ .

$x^{3} \frac{d y}{d x} + x^{3} (\frac{3}{x} y) = x^{3} (\frac{1}{5} x)$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{3}{x}$ and $y$ .

$x^{3} \frac{d y}{d x} + x^{3} \frac{3 y}{x} = x^{3} (\frac{1}{5} x)$

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

$x^{3} \frac{d y}{d x} + x \cdot x^{2} \frac{3 y}{x} = x^{3} (\frac{1}{5} x)$

Step 3.2.2.2

Cancel the common factor.

$x^{3} \frac{d y}{d x} + x \cdot x^{2} \frac{3 y}{x} = x^{3} (\frac{1}{5} x)$

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

$x^{3} \frac{d y}{d x} + 3 x^{2} y = \frac{1}{5} x^{3} x$

Step 3.4

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.4.1

Move $x$ .

$x^{3} \frac{d y}{d x} + 3 x^{2} y = \frac{1}{5} (x \cdot x^{3})$

Step 3.4.2

Multiply $x$ by $x^{3}$ .

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

$x^{3} \frac{d y}{d x} + 3 x^{2} y = \frac{1}{5} x^{1 + 3}$

Step 3.4.3

Add $1$ and $3$ .

$x^{3} \frac{d y}{d x} + 3 x^{2} y = \frac{1}{5} x^{4}$

Step 3.5

Combine $\frac{1}{5}$ and $x^{4}$ .

$x^{3} \frac{d y}{d x} + 3 x^{2} y = \frac{x^{4}}{5}$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [x^{3} y] = \frac{x^{4}}{5}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{3} y] d x = \int \frac{x^{4}}{5} d x$

Step 6

Integrate the left side.

$x^{3} y = \int \frac{x^{4}}{5} d x$

Step 7

Integrate the right side.

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Step 7.1

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$x^{3} y = \frac{1}{5} \int x^{4} d x$

Step 7.2

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

$x^{3} y = \frac{1}{5} (\frac{1}{5} x^{5} + C)$

Step 7.3

Simplify the answer.

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Step 7.3.1

Rewrite $\frac{1}{5} (\frac{1}{5} x^{5} + C)$ as $\frac{1}{5} \cdot \frac{1}{5} x^{5} + C$ .

$x^{3} y = \frac{1}{5} \cdot \frac{1}{5} x^{5} + C$

Step 7.3.2

Simplify.

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Step 7.3.2.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$x^{3} y = \frac{1}{5 \cdot 5} x^{5} + C$

Step 7.3.2.2

Multiply $5$ by $5$ .

$x^{3} y = \frac{1}{25} x^{5} + C$

Step 8

Divide each term in $x^{3} y = \frac{1}{25} x^{5} + C$ by $x^{3}$ and simplify.

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Step 8.1

Divide each term in $x^{3} y = \frac{1}{25} x^{5} + C$ by $x^{3}$ .

$\frac{x^{3} y}{x^{3}} = \frac{\frac{1}{25} x^{5}}{x^{3}} + \frac{C}{x^{3}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{x^{3} y}{x^{3}} = \frac{\frac{1}{25} x^{5}}{x^{3}} + \frac{C}{x^{3}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{25} x^{5}}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $x^{5}$ and $x^{3}$ .

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Step 8.3.1.1.1

Factor $x^{3}$ out of $\frac{1}{25} x^{5}$ .

$y = \frac{x^{3} (\frac{1}{25} x^{2})}{x^{3}} + \frac{C}{x^{3}}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

$y = \frac{x^{3} (\frac{1}{25} x^{2})}{x^{3} \cdot 1} + \frac{C}{x^{3}}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{3} (\frac{1}{25} x^{2})}{x^{3} \cdot 1} + \frac{C}{x^{3}}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{25} x^{2}}{1} + \frac{C}{x^{3}}$

Step 8.3.1.1.2.4

Divide $\frac{1}{25} x^{2}$ by $1$ .

$y = \frac{1}{25} x^{2} + \frac{C}{x^{3}}$

Step 8.3.1.2

Combine $\frac{1}{25}$ and $x^{2}$ .

$y = \frac{x^{2}}{25} + \frac{C}{x^{3}}$