Calculus Examples

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

Step 1

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

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Step 1.1

Divide each term in $(x + 1) \frac{d y}{d x} - y = e^{x} {(x + 1)}^{2}$ by $x + 1$ .

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

Step 1.2

Cancel the common factor of $x + 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 1.3.1

Factor $x + 1$ out of $e^{x} {(x + 1)}^{2}$ .

$\frac{d y}{d x} + \frac{- y}{x + 1} = \frac{(x + 1) (e^{x} (x + 1))}{x + 1}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Multiply by $1$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{- y}{x + 1} = \frac{e^{x} (x + 1)}{1}$

Step 1.3.2.4

Divide $e^{x} (x + 1)$ by $1$ .

$\frac{d y}{d x} + \frac{- y}{x + 1} = e^{x} (x + 1)$

Step 1.4

Factor $y$ out of $\frac{- y}{x + 1}$ .

$\frac{d y}{d x} + y \frac{- 1}{x + 1} = e^{x} (x + 1)$

Step 1.5

Reorder $y$ and $\frac{- 1}{x + 1}$ .

$\frac{d y}{d x} + \frac{- 1}{x + 1} y = e^{x} (x + 1)$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 1}{x + 1}$ .

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Step 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.2.3

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.3.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.2.3.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 2.2.3.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.2.3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 2.2.3.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.2.3.1.5

Add $1$ and $0$ .

$1$

Step 2.2.3.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.2.6

Replace all occurrences of $u$ with $x + 1$ .

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

${(x + 1)}^{- 1}$

Step 2.6

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

$\frac{1}{x + 1}$

Step 3

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

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Step 3.1

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

$\frac{1}{x + 1} \frac{d y}{d x} + \frac{1}{x + 1} (\frac{- 1}{x + 1} y) = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2

Simplify each term.

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Step 3.2.1

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

$\frac{\frac{d y}{d x}}{x + 1} + \frac{1}{x + 1} (\frac{- 1}{x + 1} y) = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2.2

Move the negative in front of the fraction.

$\frac{\frac{d y}{d x}}{x + 1} + \frac{1}{x + 1} (- \frac{1}{x + 1} y) = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x + 1} - \frac{1}{x + 1} (\frac{1}{x + 1} y) = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2.4

Combine $\frac{1}{x + 1}$ and $y$ .

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

Step 3.2.5

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

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Step 3.2.5.1

Multiply $\frac{y}{x + 1}$ by $\frac{1}{x + 1}$ .

$\frac{\frac{d y}{d x}}{x + 1} - \frac{y}{(x + 1) (x + 1)} = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2.5.2

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

$\frac{\frac{d y}{d x}}{x + 1} - \frac{y}{{(x + 1)}^{1} (x + 1)} = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2.5.3

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

$\frac{\frac{d y}{d x}}{x + 1} - \frac{y}{{(x + 1)}^{1} {(x + 1)}^{1}} = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2.5.4

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

$\frac{\frac{d y}{d x}}{x + 1} - \frac{y}{{(x + 1)}^{1 + 1}} = \frac{1}{x + 1} (e^{x} (x + 1))$

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

To write $\frac{\frac{d y}{d x}}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 3.4

Write each expression with a common denominator of ${(x + 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{\frac{d y}{d x}}{x + 1}$ by $\frac{x + 1}{x + 1}$ .

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

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

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

Step 3.7

Cancel the common factor of $x + 1$ .

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Step 3.7.1

Factor $x + 1$ out of $e^{x} (x + 1)$ .

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

Step 3.7.2

Cancel the common factor.

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

Step 3.7.3

Rewrite the expression.

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

Step 3.8

Reorder factors in $\frac{\frac{d y}{d x} x + \frac{d y}{d x} - y}{{(x + 1)}^{2}} = e^{x}$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

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

Step 8

Solve for $y$ .

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Step 8.1

Combine $\frac{1}{x + 1}$ and $y$ .

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

Step 8.2

Multiply both sides by $x + 1$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Cancel the common factor of $x + 1$ .

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Rewrite the expression.

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

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(e^{x} + C) (x + 1)$ .

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Step 8.3.2.1.1

Expand $(e^{x} + C) (x + 1)$ using the FOIL Method.

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Step 8.3.2.1.1.1

Apply the distributive property.

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

Step 8.3.2.1.1.2

Apply the distributive property.

$y = e^{x} x + e^{x} \cdot 1 + C (x + 1)$

Step 8.3.2.1.1.3

Apply the distributive property.

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

Step 8.3.2.1.2

Simplify terms.

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Step 8.3.2.1.2.1

Simplify each term.

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Step 8.3.2.1.2.1.1

Multiply $e^{x}$ by $1$ .

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

Step 8.3.2.1.2.1.2

Multiply $C$ by $1$ .

$y = e^{x} x + e^{x} + C x + C$

Step 8.3.2.1.2.2

Simplify the expression.

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Step 8.3.2.1.2.2.1

Reorder factors in $e^{x} x + e^{x} + C x + C$ .

$y = x e^{x} + e^{x} + C x + C$

Step 8.3.2.1.2.2.2

Move $e^{x}$ .

$y = x e^{x} + C x + C + e^{x}$

Step 8.3.2.1.2.2.3

Move $x e^{x}$ .

$y = C x + C + x e^{x} + e^{x}$