Calculus Examples

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

Step 1

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

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Step 1.1

Reorder terms.

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

Step 1.2

Factor $y$ out of $- \frac{2 y}{x}$ .

$\frac{d y}{d x} + y (- \frac{2}{x}) = 1 + \sqrt{x}$

Step 1.3

Reorder $y$ and $- \frac{2}{x}$ .

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

Step 2

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 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $- 1$ .

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 2}$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{2}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{2}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $- \frac{1}{x^{2}} \cdot \frac{2 y}{x}$ .

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Step 3.2.4.1

Multiply $\frac{2 y}{x}$ by $\frac{1}{x^{2}}$ .

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

Step 3.2.4.2

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

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Step 3.2.4.2.1

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

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Step 3.2.4.2.1.1

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

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

Step 3.2.4.2.1.2

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

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

Step 3.2.4.2.2

Add $1$ and $2$ .

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

Step 3.3

Simplify each term.

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Step 3.3.1

Multiply $\frac{1}{x^{2}}$ by $1$ .

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

Step 3.3.2

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{2}} y] d x = \int \frac{1}{x^{2}} + \frac{\sqrt{x}}{x^{2}} d x$

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{x^{2}} y = \int {(x^{2})}^{- 1} d x + \int \frac{\sqrt{x}}{x^{2}} d x$

Step 7.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.2.2.2

Multiply $2$ by $- 1$ .

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

Step 7.3

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$\frac{1}{x^{2}} y = - x^{- 1} + C + \int \frac{\sqrt{x}}{x^{2}} d x$

Step 7.4

Simplify the expression.

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Step 7.4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{x^{2}} y = - x^{- 1} + C + \int \sqrt{x} {(x^{2})}^{- 1} d x$

Step 7.4.2

Simplify.

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Step 7.4.2.1

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.4.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{x^{2}} y = - x^{- 1} + C + \int \sqrt{x} x^{2 \cdot - 1} d x$

Step 7.4.2.1.2

Multiply $2$ by $- 1$ .

$\frac{1}{x^{2}} y = - x^{- 1} + C + \int \sqrt{x} x^{- 2} d x$

Step 7.4.2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 7.4.2.3

Multiply $x^{\frac{1}{2}}$ by $x^{- 2}$ by adding the exponents.

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Step 7.4.2.3.1

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

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

Step 7.4.2.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.4.2.3.3

Combine $- 2$ and $\frac{2}{2}$ .

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

Step 7.4.2.3.4

Combine the numerators over the common denominator.

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

Step 7.4.2.3.5

Simplify the numerator.

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Step 7.4.2.3.5.1

Multiply $- 2$ by $2$ .

$\frac{1}{x^{2}} y = - x^{- 1} + C + \int x^{\frac{1 - 4}{2}} d x$

Step 7.4.2.3.5.2

Subtract $4$ from $1$ .

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

Step 7.4.2.3.6

Move the negative in front of the fraction.

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

Step 7.5

By the Power Rule, the integral of $x^{- \frac{3}{2}}$ with respect to $x$ is $- 2 x^{- \frac{1}{2}}$ .

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

Step 7.6

Simplify.

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

Step 8

Solve for $y$ .

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Step 8.1

Move all terms containing variables to the left side of the equation.

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Step 8.1.1

Add $\frac{1}{x}$ to both sides of the equation.

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

Step 8.1.2

Add $2 x^{- \frac{1}{2}}$ to both sides of the equation.

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

Step 8.1.3

Subtract $C$ from both sides of the equation.

$\frac{1}{x^{2}} y + \frac{1}{x} + 2 x^{- \frac{1}{2}} - C = 0$

Step 8.1.4

Simplify each term.

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Step 8.1.4.1

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

$\frac{y}{x^{2}} + \frac{1}{x} + 2 x^{- \frac{1}{2}} - C = 0$

Step 8.1.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{y}{x^{2}} + \frac{1}{x} + 2 \frac{1}{x^{\frac{1}{2}}} - C = 0$

Step 8.1.4.3

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

$\frac{y}{x^{2}} + \frac{1}{x} + \frac{2}{x^{\frac{1}{2}}} - C = 0$

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Subtract $\frac{2}{x^{\frac{1}{2}}}$ from both sides of the equation.

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

Step 8.2.3

Add $C$ to both sides of the equation.

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

Step 8.3

Multiply both sides by $x^{2}$ .

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

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

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

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(- \frac{1}{x} - \frac{2}{x^{\frac{1}{2}}} + C) x^{2}$ .

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

Simplify.

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Step 8.4.2.1.2.1

Cancel the common factor of $x$ .

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Step 8.4.2.1.2.1.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 8.4.2.1.2.1.2

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

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

Step 8.4.2.1.2.1.3

Cancel the common factor.

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

Step 8.4.2.1.2.1.4

Rewrite the expression.

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

Step 8.4.2.1.2.2

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 8.4.2.1.2.2.1

Move the leading negative in $- \frac{2}{x^{\frac{1}{2}}}$ into the numerator.

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

Step 8.4.2.1.2.2.2

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

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

Step 8.4.2.1.2.2.3

Cancel the common factor.

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

Step 8.4.2.1.2.2.4

Rewrite the expression.

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

Step 8.4.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 8.4.2.1.4

Move $- 2 x^{\frac{3}{2}}$ .

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

Step 8.4.2.1.5

Reorder $- x$ and $C x^{2}$ .

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