Calculus Examples

$x \frac{d y}{d x} + 5 y - x^{2} = 0$

Step 1

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

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Step 1.1

Add $x^{2}$ to both sides of the equation.

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

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

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Step 1.4.1

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

$\frac{d y}{d x} + \frac{5 y}{x} = \frac{x \cdot x}{x}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Cancel the common factor.

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

Step 1.4.2.4

Rewrite the expression.

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

Step 1.4.2.5

Divide $x$ by $1$ .

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

Step 1.5

Factor $y$ out of $\frac{5 y}{x}$ .

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

Step 1.6

Reorder $y$ and $\frac{5}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{5}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{5}{x} d x}$

Step 2.2

Integrate $\frac{5}{x}$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$e^{5 \int \frac{1}{x} d x}$

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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

Step 2.3

Remove the constant of integration.

$e^{5 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{5})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{5}$

Step 3

Multiply each term by the integrating factor $x^{5}$ .

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Step 3.1

Multiply each term by $x^{5}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{5}{x}$ and $y$ .

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

$x^{5} \frac{d y}{d x} + x^{4} (5 y) = x^{5} x$

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.3

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 3.3.1

Multiply $x^{5}$ by $x$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.2

Add $5$ and $1$ .

$x^{5} \frac{d y}{d x} + 5 x^{4} y = x^{6}$

Step 4

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

$\frac{d}{d x} [x^{5} y] = x^{6}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{5} y] d x = \int x^{6} d x$

Step 6

Integrate the left side.

$x^{5} y = \int x^{6} d x$

Step 7

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$x^{5} y = \frac{1}{7} x^{7} + C$

Step 8

Divide each term in $x^{5} y = \frac{1}{7} x^{7} + C$ by $x^{5}$ and simplify.

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Step 8.1

Divide each term in $x^{5} y = \frac{1}{7} x^{7} + C$ by $x^{5}$ .

$\frac{x^{5} y}{x^{5}} = \frac{\frac{1}{7} x^{7}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{x^{5} y}{x^{5}} = \frac{\frac{1}{7} x^{7}}{x^{5}} + \frac{C}{x^{5}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{7} x^{7}}{x^{5}} + \frac{C}{x^{5}}$

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 $x^{7}$ and $x^{5}$ .

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Step 8.3.1.1.1

Factor $x^{5}$ out of $\frac{1}{7} x^{7}$ .

$y = \frac{x^{5} (\frac{1}{7} x^{2})}{x^{5}} + \frac{C}{x^{5}}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

$y = \frac{x^{5} (\frac{1}{7} x^{2})}{x^{5} \cdot 1} + \frac{C}{x^{5}}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{5} (\frac{1}{7} x^{2})}{x^{5} \cdot 1} + \frac{C}{x^{5}}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{7} x^{2}}{1} + \frac{C}{x^{5}}$

Step 8.3.1.1.2.4

Divide $\frac{1}{7} x^{2}$ by $1$ .

$y = \frac{1}{7} x^{2} + \frac{C}{x^{5}}$

Step 8.3.1.2

Combine $\frac{1}{7}$ and $x^{2}$ .

$y = \frac{x^{2}}{7} + \frac{C}{x^{5}}$