Calculus Examples

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

Step 1

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

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Step 1.1

Factor $y$ out of $- \frac{2 y}{x + 1}$ .

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

Step 1.2

Reorder $y$ and $- \frac{2}{x + 1}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - \frac{2}{x + 1}$ .

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $- \frac{2}{x + 1}$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $- 1$ .

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

Step 2.2.4

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

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Step 2.2.4.1

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

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Step 2.2.4.1.1

Differentiate $x + 1$ .

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

Step 2.2.4.1.2

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 2.2.4.1.3

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 2.2.4.1.4

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

$1 + 0$

Step 2.2.4.1.5

Add $1$ and $0$ .

$1$

Step 2.2.4.2

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.2.7

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

$e^{- 2 \ln (| x + 1 |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 2 \ln (x + 1)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln ({(x + 1)}^{- 2})}$

Step 2.5

Exponentiation and log are inverse functions.

${(x + 1)}^{- 2}$

Step 2.6

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

$\frac{1}{{(x + 1)}^{2}}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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

Step 3.2.4

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

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Step 3.2.4.1

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

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

Step 3.2.4.2

Multiply $x + 1$ by ${(x + 1)}^{2}$ by adding the exponents.

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Step 3.2.4.2.1

Multiply $x + 1$ by ${(x + 1)}^{2}$ .

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Step 3.2.4.2.1.1

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

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

Step 3.2.4.2.1.2

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

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

Step 3.2.4.2.2

Add $1$ and $2$ .

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

Step 3.3

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

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Step 3.3.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{3}$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

Apply the constant rule.

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

Step 7.4

Simplify.

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Multiply both sides by ${(x + 1)}^{2}$ .

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

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 ${(x + 1)}^{2}$ .

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Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

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

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Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

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

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

Step 8.4.2.1.3

To write $x {(x + 1)}^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.4.2.1.4

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

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

Step 8.4.2.1.5

Combine the numerators over the common denominator.

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

Step 8.4.2.1.6

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

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Step 8.4.2.1.6.1

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

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

Step 8.4.2.1.6.2

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

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

Step 8.4.2.1.6.3

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

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

Step 8.4.2.1.7

To write $C {(x + 1)}^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.4.2.1.8

Simplify terms.

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Step 8.4.2.1.8.1

Combine $C {(x + 1)}^{2}$ and $\frac{2}{2}$ .

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

Step 8.4.2.1.8.2

Combine the numerators over the common denominator.

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

Step 8.4.2.1.9

Simplify the numerator.

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Step 8.4.2.1.9.1

Factor ${(x + 1)}^{2}$ out of $x {(x + 1)}^{2} (x + 2) + C {(x + 1)}^{2} \cdot 2$ .

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Step 8.4.2.1.9.1.1

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

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

Step 8.4.2.1.9.1.2

Factor ${(x + 1)}^{2}$ out of $C {(x + 1)}^{2} \cdot 2$ .

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

Step 8.4.2.1.9.1.3

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{2} (x (x + 2)) + {(x + 1)}^{2} (C \cdot 2)$ .

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

Step 8.4.2.1.9.2

Apply the distributive property.

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

Step 8.4.2.1.9.3

Multiply $x$ by $x$ .

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

Step 8.4.2.1.9.4

Move $2$ to the left of $x$ .

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

Step 8.4.2.1.9.5

Move $2$ to the left of $C$ .

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