Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{2}{x}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{2}{x} d x}$

Step 1.2

Integrate $\frac{2}{x}$ .

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Step 1.2.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{2 \int \frac{1}{x} d x}$

Step 1.2.2

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

$e^{2 (\ln (| x |) + C)}$

Step 1.2.3

Simplify.

$e^{2 \ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{2 \ln (x)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (x^{2})}$

Step 1.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 2

Multiply each term by the integrating factor $x^{2}$ .

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Step 2.1

Multiply each term by $x^{2}$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{2}{x}$ and $y$ .

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

Step 2.2.2

Cancel the common factor of $x$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Rewrite using the commutative property of multiplication.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Add $2$ and $2$ .

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

Step 3

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

$\frac{d}{d x} [x^{2} y] = x^{4}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int x^{4} d x$

Step 5

Integrate the left side.

$x^{2} y = \int x^{4} d x$

Step 6

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

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

Step 7

Divide each term in $x^{2} y = \frac{1}{5} x^{5} + C$ by $x^{2}$ and simplify.

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $x^{5}$ and $x^{2}$ .

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Step 7.3.1.1.1

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

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

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

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

Step 7.3.1.1.2.2

Cancel the common factor.

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

Step 7.3.1.1.2.3

Rewrite the expression.

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

Step 7.3.1.1.2.4

Divide $\frac{1}{5} x^{3}$ by $1$ .

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

Step 7.3.1.2

Combine $\frac{1}{5}$ and $x^{3}$ .

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

Step 8

Use the initial condition to find the value of $C$ by substituting $1$ for $x$ and $\frac{2}{5}$ for $y$ in $y = \frac{x^{3}}{5} + \frac{C}{x^{2}}$ .

$\frac{2}{5} = \frac{1^{3}}{5} + \frac{C}{1^{2}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $\frac{1^{3}}{5} + \frac{C}{1^{2}} = \frac{2}{5}$ .

$\frac{1^{3}}{5} + \frac{C}{1^{2}} = \frac{2}{5}$

Step 9.2

Simplify each term.

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Step 9.2.1

One to any power is one.

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

Step 9.2.2

One to any power is one.

$\frac{1}{5} + \frac{C}{1} = \frac{2}{5}$

Step 9.2.3

Divide $C$ by $1$ .

$\frac{1}{5} + C = \frac{2}{5}$

Step 9.3

Move all terms not containing $C$ to the right side of the equation.

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Step 9.3.1

Subtract $\frac{1}{5}$ from both sides of the equation.

$C = \frac{2}{5} - \frac{1}{5}$

Step 9.3.2

Combine the numerators over the common denominator.

$C = \frac{2 - 1}{5}$

Step 9.3.3

Subtract $1$ from $2$ .

$C = \frac{1}{5}$

Step 10

Substitute $\frac{1}{5}$ for $C$ in $y = \frac{x^{3}}{5} + \frac{C}{x^{2}}$ and simplify.

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Step 10.1

Substitute $\frac{1}{5}$ for $C$ .

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

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 10.2.2

Combine.

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

Step 10.2.3

Multiply $1$ by $1$ .

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