Calculus Examples

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

Step 1

To solve the differential equation, let

$v = y^{1 - n}$ where

$n$ is the exponent of

$y^{2}$ .

$v = y^{- 1}$

Step 2

Solve the equation for

$y$ .

$y = v^{- 1}$

Step 3

Take the derivative of

$y$ with respect to

$x$ .

$y' = v^{- 1}$

Step 4

Take the derivative of

$v^{- 1}$ with respect to

$x$ .

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Step 4.1

Take the derivative of

$v^{- 1}$ .

$y' = \frac{d}{d x} [v^{- 1}]$

Step 4.2

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$y' = \frac{d}{d x} [\frac{1}{v}]$

Step 4.3

Differentiate using the Quotient Rule which states that

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

$f (x) = 1$ and

$g (x) = v$ .

$y' = \frac{v \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v]}{v^{2}}$

Step 4.4

Differentiate using the Constant Rule.

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Step 4.4.1

Multiply

$- 1$ by

$1$ .

$y' = \frac{v \frac{d}{d x} [1] - \frac{d}{d x} [v]}{v^{2}}$

Step 4.4.2

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$y' = \frac{v \cdot 0 - \frac{d}{d x} [v]}{v^{2}}$

Step 4.4.3

Simplify the expression.

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Step 4.4.3.1

Multiply

$v$ by

$0$ .

$y' = \frac{0 - \frac{d}{d x} [v]}{v^{2}}$

Step 4.4.3.2

Subtract

$\frac{d}{d x} [v]$ from

$0$ .

$y' = \frac{- \frac{d}{d x} [v]}{v^{2}}$

Step 4.4.3.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v]}{v^{2}}$

Step 4.5

Rewrite

$\frac{d}{d x} [v]$ as

$v'$ .

$y' = - \frac{v'}{v^{2}}$

Step 5

Substitute

$- \frac{v'}{v^{2}}$ for

$\frac{d y}{d x}$ and

$v^{- 1}$ for

$y$ in the original equation

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

$- \frac{v'}{v^{2}} + \frac{1}{x} v^{- 1} = x^{4} {(v^{- 1})}^{2}$

Step 6

Solve the substituted differential equation.

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Step 6.1

Rewrite the differential equation as

$\frac{d v}{d x} + M (x) v = Q (x)$ .

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Step 6.1.1

Multiply each term in

$- \frac{\frac{d v}{d x}}{v^{2}} + \frac{1}{x} v^{- 1} = x^{4} {(v^{- 1})}^{2}$ by

$- v^{2}$ to eliminate the fractions.

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Step 6.1.1.1

Multiply each term in

$- \frac{\frac{d v}{d x}}{v^{2}} + \frac{1}{x} v^{- 1} = x^{4} {(v^{- 1})}^{2}$ by

$- v^{2}$ .

$- \frac{\frac{d v}{d x}}{v^{2}} (- v^{2}) + \frac{1}{x} v^{- 1} (- v^{2}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2

Simplify the left side.

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Step 6.1.1.2.1

Simplify each term.

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Step 6.1.1.2.1.1

Cancel the common factor of

$v^{2}$ .

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Step 6.1.1.2.1.1.1

Move the leading negative in

$- \frac{\frac{d v}{d x}}{v^{2}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{v^{2}} (- v^{2}) + \frac{1}{x} v^{- 1} (- v^{2}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.1.2

Factor

$v^{2}$ out of

$- v^{2}$ .

$\frac{- \frac{d v}{d x}}{v^{2}} (v^{2} \cdot - 1) + \frac{1}{x} v^{- 1} (- v^{2}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{v^{2}} (v^{2} \cdot - 1) + \frac{1}{x} v^{- 1} (- v^{2}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 + \frac{1}{x} v^{- 1} (- v^{2}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.2

Multiply

$- 1$ by

$- 1$ .

$1 \frac{d v}{d x} + \frac{1}{x} v^{- 1} (- v^{2}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.3

Multiply

$\frac{d v}{d x}$ by

$1$ .

$\frac{d v}{d x} + \frac{1}{x} v^{- 1} (- v^{2}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - \frac{1}{x} v^{- 1} v^{2} = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.5

Multiply

$v^{- 1}$ by

$v^{2}$ by adding the exponents.

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Step 6.1.1.2.1.5.1

Move

$v^{2}$ .

$\frac{d v}{d x} - \frac{1}{x} (v^{2} v^{- 1}) = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.5.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d v}{d x} - \frac{1}{x} v^{2 - 1} = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.5.3

Subtract

$1$ from

$2$ .

$\frac{d v}{d x} - \frac{1}{x} v^{1} = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.6

Simplify

$- \frac{1}{x} v^{1}$ .

$\frac{d v}{d x} - \frac{1}{x} v = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.2.1.7

Combine

$v$ and

$\frac{1}{x}$ .

$\frac{d v}{d x} - \frac{v}{x} = x^{4} {(v^{- 1})}^{2} (- v^{2})$

Step 6.1.1.3

Simplify the right side.

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Step 6.1.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - \frac{v}{x} = - x^{4} {(v^{- 1})}^{2} v^{2}$

Step 6.1.1.3.2

Multiply the exponents in

${(v^{- 1})}^{2}$ .

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Step 6.1.1.3.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{d v}{d x} - \frac{v}{x} = - x^{4} v^{- 1 \cdot 2} v^{2}$

Step 6.1.1.3.2.2

Multiply

$- 1$ by

$2$ .

$\frac{d v}{d x} - \frac{v}{x} = - x^{4} v^{- 2} v^{2}$

Step 6.1.1.3.3

Multiply

$v^{- 2}$ by

$v^{2}$ by adding the exponents.

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Step 6.1.1.3.3.1

Move

$v^{2}$ .

$\frac{d v}{d x} - \frac{v}{x} = - x^{4} (v^{2} v^{- 2})$

Step 6.1.1.3.3.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d v}{d x} - \frac{v}{x} = - x^{4} v^{2 - 2}$

Step 6.1.1.3.3.3

Subtract

$2$ from

$2$ .

$\frac{d v}{d x} - \frac{v}{x} = - x^{4} v^{0}$

Step 6.1.1.3.4

Simplify

$- x^{4} v^{0}$ .

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

Step 6.1.2

Factor

$v$ out of

$- \frac{v}{x}$ .

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

Step 6.1.3

Reorder

$v$ and

$- \frac{1}{x}$ .

$\frac{d v}{d x} - \frac{1}{x} v = - x^{4}$

Step 6.2

The integrating factor is defined by the formula

$e^{\int P (x) d x}$ , where

$P (x) = - \frac{1}{x}$ .

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Step 6.2.1

Set up the integration.

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

Step 6.2.2

Integrate

$- \frac{1}{x}$ .

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Step 6.2.2.1

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

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

Step 6.2.2.2

The integral of

$\frac{1}{x}$ with respect to

$x$ is

$\ln (| x |)$ .

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

Step 6.2.2.3

Simplify.

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

Step 6.2.3

Remove the constant of integration.

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

Step 6.2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 1})}$

Step 6.2.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 6.2.6

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x}$

Step 6.3

Multiply each term by the integrating factor

$\frac{1}{x}$ .

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Step 6.3.1

Multiply each term by

$\frac{1}{x}$ .

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

Step 6.3.2

Simplify each term.

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Step 6.3.2.1

Combine

$\frac{1}{x}$ and

$\frac{d v}{d x}$ .

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

Step 6.3.2.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{d v}{d x}}{x} - \frac{1}{x} (\frac{1}{x} v) = \frac{1}{x} (- x^{4})$

Step 6.3.2.3

Combine

$\frac{1}{x}$ and

$v$ .

$\frac{\frac{d v}{d x}}{x} - \frac{1}{x} \cdot \frac{v}{x} = \frac{1}{x} (- x^{4})$

Step 6.3.2.4

Multiply

$- \frac{1}{x} \cdot \frac{v}{x}$ .

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Step 6.3.2.4.1

Multiply

$\frac{v}{x}$ by

$\frac{1}{x}$ .

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x \cdot x} = \frac{1}{x} (- x^{4})$

Step 6.3.2.4.2

Raise

$x$ to the power of

$1$ .

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{1} x} = \frac{1}{x} (- x^{4})$

Step 6.3.2.4.3

Raise

$x$ to the power of

$1$ .

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{1} x^{1}} = \frac{1}{x} (- x^{4})$

Step 6.3.2.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.3.2.4.5

Add

$1$ and

$1$ .

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = \frac{1}{x} (- x^{4})$

Step 6.3.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = - \frac{1}{x} x^{4}$

Step 6.3.4

Cancel the common factor of

$x$ .

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Step 6.3.4.1

Move the leading negative in

$- \frac{1}{x}$ into the numerator.

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = \frac{- 1}{x} x^{4}$

Step 6.3.4.2

Factor

$x$ out of

$x^{4}$ .

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = \frac{- 1}{x} (x \cdot x^{3})$

Step 6.3.4.3

Cancel the common factor.

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = \frac{- 1}{x} (x \cdot x^{3})$

Step 6.3.4.4

Rewrite the expression.

$\frac{\frac{d v}{d x}}{x} - \frac{v}{x^{2}} = - x^{3}$

Step 6.4

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

$\frac{d}{d x} [\frac{1}{x} v] = - x^{3}$

Step 6.5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x} v] d x = \int - x^{3} d x$

Step 6.6

Integrate the left side.

$\frac{1}{x} v = \int - x^{3} d x$

Step 6.7

Integrate the right side.

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Step 6.7.1

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

$\frac{1}{x} v = - \int x^{3} d x$

Step 6.7.2

By the Power Rule, the integral of

$x^{3}$ with respect to

$x$ is

$\frac{1}{4} x^{4}$ .

$\frac{1}{x} v = - (\frac{1}{4} x^{4} + C)$

Step 6.7.3

Rewrite

$- (\frac{1}{4} x^{4} + C)$ as

$- \frac{1}{4} x^{4} + C$ .

$\frac{1}{x} v = - \frac{1}{4} x^{4} + C$

Step 6.8

Solve for

$v$ .

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Step 6.8.1

Combine

$\frac{1}{x}$ and

$v$ .

$\frac{v}{x} = - \frac{1}{4} x^{4} + C$

Step 6.8.2

Combine

$x^{4}$ and

$\frac{1}{4}$ .

$\frac{v}{x} = - \frac{x^{4}}{4} + C$

Step 6.8.3

Multiply both sides by

$x$ .

$\frac{v}{x} x = (- \frac{x^{4}}{4} + C) x$

Step 6.8.4

Simplify.

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Step 6.8.4.1

Simplify the left side.

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Step 6.8.4.1.1

Cancel the common factor of

$x$ .

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Step 6.8.4.1.1.1

Cancel the common factor.

$\frac{v}{x} x = (- \frac{x^{4}}{4} + C) x$

Step 6.8.4.1.1.2

Rewrite the expression.

$v = (- \frac{x^{4}}{4} + C) x$

Step 6.8.4.2

Simplify the right side.

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Step 6.8.4.2.1

Simplify

$(- \frac{x^{4}}{4} + C) x$ .

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Step 6.8.4.2.1.1

Apply the distributive property.

$v = - \frac{x^{4}}{4} x + C x$

Step 6.8.4.2.1.2

Multiply

$- \frac{x^{4}}{4} x$ .

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Step 6.8.4.2.1.2.1

Combine

$x$ and

$\frac{x^{4}}{4}$ .

$v = - \frac{x \cdot x^{4}}{4} + C x$

Step 6.8.4.2.1.2.2

Multiply

$x$ by

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

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Step 6.8.4.2.1.2.2.1

Multiply

$x$ by

$x^{4}$ .

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Step 6.8.4.2.1.2.2.1.1

Raise

$x$ to the power of

$1$ .

$v = - \frac{x^{1} x^{4}}{4} + C x$

Step 6.8.4.2.1.2.2.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$v = - \frac{x^{1 + 4}}{4} + C x$

Step 6.8.4.2.1.2.2.2

Add

$1$ and

$4$ .

$v = - \frac{x^{5}}{4} + C x$

Step 7

Substitute

$y^{- 1}$ for

$v$ .

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

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