Calculus Examples

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

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

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

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

Step 6

Solve the substituted differential equation.

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Step 6.1

Multiply each term in

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

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

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Step 6.1.1

Multiply each term in

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

$- v^{2}$ .

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

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Cancel the common factor of

$v^{2}$ .

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Step 6.1.2.1.1.1

Move the leading negative in

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

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

Step 6.1.2.1.1.2

Factor

$v^{2}$ out of

$- v^{2}$ .

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

Step 6.1.2.1.1.3

Cancel the common factor.

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

Step 6.1.2.1.1.4

Rewrite the expression.

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

Step 6.1.2.1.2

Multiply

$- 1$ by

$- 1$ .

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

Step 6.1.2.1.3

Multiply

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

$1$ .

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

Step 6.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 6.1.2.1.5

Multiply

$v^{- 1}$ by

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

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Step 6.1.2.1.5.1

Move

$v^{2}$ .

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

Step 6.1.2.1.5.2

Use the power rule

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

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

Step 6.1.2.1.5.3

Subtract

$1$ from

$2$ .

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

Step 6.1.2.1.6

Simplify

$- v^{1}$ .

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

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.3.2

Multiply the exponents in

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

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Step 6.1.3.2.1

Apply the power rule and multiply exponents,

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

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

Step 6.1.3.2.2

Multiply

$- 1$ by

$2$ .

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

Step 6.1.3.3

Multiply

$v^{- 2}$ by

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

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Step 6.1.3.3.1

Move

$v^{2}$ .

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

Step 6.1.3.3.2

Use the power rule

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

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

Step 6.1.3.3.3

Subtract

$2$ from

$2$ .

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

Step 6.1.3.4

Simplify

$- x v^{0}$ .

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

Step 6.2

The integrating factor is defined by the formula

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

$P (x) = - 1$ .

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Step 6.2.1

Set up the integration.

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

Step 6.2.2

Apply the constant rule.

$e^{- x + C}$

Step 6.2.3

Remove the constant of integration.

$e^{- x}$

Step 6.3

Multiply each term by the integrating factor

$e^{- x}$ .

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Step 6.3.1

Multiply each term by

$e^{- x}$ .

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

Step 6.3.2

Rewrite using the commutative property of multiplication.

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

Step 6.3.3

Rewrite using the commutative property of multiplication.

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

Step 6.3.4

Reorder factors in

$e^{- x} \frac{d v}{d x} - e^{- x} v = - e^{- x} x$ .

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

Step 6.4

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

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

Step 6.5

Set up an integral on each side.

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

Step 6.6

Integrate the left side.

$e^{- x} v = \int - x e^{- x} d x$

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.

$e^{- x} v = - \int x e^{- x} d x$

Step 6.7.2

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = x$ and

$d v = e^{- x}$ .

$e^{- x} v = - (x (- e^{- x}) - \int - e^{- x} d x)$

Step 6.7.3

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

$e^{- x} v = - (x (- e^{- x}) - - \int e^{- x} d x)$

Step 6.7.4

Simplify.

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Step 6.7.4.1

Multiply

$- 1$ by

$- 1$ .

$e^{- x} v = - (x (- e^{- x}) + 1 \int e^{- x} d x)$

Step 6.7.4.2

Multiply

$\int e^{- x} d x$ by

$1$ .

$e^{- x} v = - (x (- e^{- x}) + \int e^{- x} d x)$

Step 6.7.5

Let

$u = - x$ . Then

$d u = - d x$ , so

$- d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

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Step 6.7.5.1

Let

$u = - x$ . Find

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

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Step 6.7.5.1.1

Differentiate

$- x$ .

$\frac{d}{d x} [- x]$

Step 6.7.5.1.2

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x$ with respect to

$x$ is

$- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 6.7.5.1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$- 1 \cdot 1$

Step 6.7.5.1.4

Multiply

$- 1$ by

$1$ .

$- 1$

Step 6.7.5.2

Rewrite the problem using

$u$ and

$d u$ .

$e^{- x} v = - (x (- e^{- x}) + \int - e^{u} d u)$

Step 6.7.6

Since

$- 1$ is constant with respect to

$u$ , move

$- 1$ out of the integral.

$e^{- x} v = - (x (- e^{- x}) - \int e^{u} d u)$

Step 6.7.7

The integral of

$e^{u}$ with respect to

$u$ is

$e^{u}$ .

$e^{- x} v = - (x (- e^{- x}) - (e^{u} + C))$

Step 6.7.8

Rewrite

$- (x (- e^{- x}) - (e^{u} + C))$ as

$- (- x e^{- x} - e^{u}) + C$ .

$e^{- x} v = - (- x e^{- x} - e^{u}) + C$

Step 6.7.9

Replace all occurrences of

$u$ with

$- x$ .

$e^{- x} v = - (- x e^{- x} - e^{- x}) + C$

Step 6.7.10

Simplify.

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Step 6.7.10.1

Apply the distributive property.

$e^{- x} v = - (- x e^{- x}) - - e^{- x} + C$

Step 6.7.10.2

Multiply

$- (- x e^{- x})$ .

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Step 6.7.10.2.1

Multiply

$- 1$ by

$- 1$ .

$e^{- x} v = 1 (x e^{- x}) - - e^{- x} + C$

Step 6.7.10.2.2

Multiply

$x$ by

$1$ .

$e^{- x} v = x e^{- x} - - e^{- x} + C$

Step 6.7.10.3

Multiply

$- - e^{- x}$ .

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Step 6.7.10.3.1

Multiply

$- 1$ by

$- 1$ .

$e^{- x} v = x e^{- x} + 1 e^{- x} + C$

Step 6.7.10.3.2

Multiply

$e^{- x}$ by

$1$ .

$e^{- x} v = x e^{- x} + e^{- x} + C$

Step 6.8

Divide each term in

$e^{- x} v = x e^{- x} + e^{- x} + C$ by

$e^{- x}$ and simplify.

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Step 6.8.1

Divide each term in

$e^{- x} v = x e^{- x} + e^{- x} + C$ by

$e^{- x}$ .

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

Step 6.8.2

Simplify the left side.

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Step 6.8.2.1

Cancel the common factor of

$e^{- x}$ .

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Step 6.8.2.1.1

Cancel the common factor.

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

Step 6.8.2.1.2

Divide

$v$ by

$1$ .

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

Step 6.8.3

Simplify the right side.

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Step 6.8.3.1

Simplify each term.

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Step 6.8.3.1.1

Cancel the common factor of

$e^{- x}$ .

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Step 6.8.3.1.1.1

Cancel the common factor.

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

Step 6.8.3.1.1.2

Divide

$x$ by

$1$ .

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

Step 6.8.3.1.2

Cancel the common factor of

$e^{- x}$ .

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Step 6.8.3.1.2.1

Cancel the common factor.

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

Step 6.8.3.1.2.2

Rewrite the expression.

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

Step 7

Substitute

$y^{- 1}$ for

$v$ .

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