Calculus Examples

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

Step 1

Let $v = \frac{d y}{d x}$ . Then $\frac{d v}{d x} = \frac{d^{2} y}{d x^{2}}$ . Substitute $v$ for $\frac{d y}{d x}$ and $\frac{d v}{d x}$ for $\frac{d^{2} y}{d x^{2}}$ to get a differential equation with dependent variable $v$ and independent variable $x$ .

$\frac{d v}{d x} + 2 v = 4 x$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int 2 d x}$

Step 2.2

Apply the constant rule.

$e^{2 x + C_{1}}$

Step 2.3

Remove the constant of integration.

$e^{2 x}$

Step 3

Multiply each term by the integrating factor $e^{2 x}$ .

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Step 3.1

Multiply each term by $e^{2 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Reorder factors in $e^{2 x} \frac{d v}{d x} + 2 e^{2 x} v = 4 e^{2 x} x$ .

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

Step 4

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

$\frac{d}{d x} [e^{2 x} v] = 4 x e^{2 x}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{2 x} v] d x = \int 4 x e^{2 x} d x$

Step 6

Integrate the left side.

$e^{2 x} v = \int 4 x e^{2 x} d x$

Step 7

Integrate the right side.

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Step 7.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$e^{2 x} v = 4 \int x e^{2 x} d x$

Step 7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

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

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

Step 7.4

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.5

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.5.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 7.5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 7.5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 7.5.1.3

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

$2 \cdot 1$

Step 7.5.1.4

Multiply $2$ by $1$ .

$2$

Step 7.5.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 7.6

Combine $e^{u_{1}}$ and $\frac{1}{2}$ .

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

Step 7.7

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.8

Simplify.

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Step 7.8.1

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

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

Step 7.8.2

Multiply $2$ by $2$ .

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

Step 7.9

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

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

Step 7.10

Rewrite $4 (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u_{1}} + C_{2}))$ as $4 (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u_{1}}) + C_{2}$ .

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

Step 7.11

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 7.12

Simplify.

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Step 7.12.1

Simplify each term.

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Step 7.12.1.1

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

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

Step 7.12.1.2

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

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

Step 7.12.1.3

Combine $e^{2 x}$ and $\frac{1}{4}$ .

$e^{2 x} v = 4 (\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4}) + C_{2}$

Step 7.12.2

Apply the distributive property.

$e^{2 x} v = 4 \frac{x e^{2 x}}{2} + 4 (- \frac{e^{2 x}}{4}) + C_{2}$

Step 7.12.3

Cancel the common factor of $2$ .

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Step 7.12.3.1

Factor $2$ out of $4$ .

$e^{2 x} v = 2 (2) \frac{x e^{2 x}}{2} + 4 (- \frac{e^{2 x}}{4}) + C_{2}$

Step 7.12.3.2

Cancel the common factor.

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

Step 7.12.3.3

Rewrite the expression.

$e^{2 x} v = 2 (x e^{2 x}) + 4 (- \frac{e^{2 x}}{4}) + C_{2}$

Step 7.12.4

Cancel the common factor of $4$ .

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Step 7.12.4.1

Move the leading negative in $- \frac{e^{2 x}}{4}$ into the numerator.

$e^{2 x} v = 2 (x e^{2 x}) + 4 \frac{- e^{2 x}}{4} + C_{2}$

Step 7.12.4.2

Cancel the common factor.

$e^{2 x} v = 2 (x e^{2 x}) + 4 \frac{- e^{2 x}}{4} + C_{2}$

Step 7.12.4.3

Rewrite the expression.

$e^{2 x} v = 2 (x e^{2 x}) - e^{2 x} + C_{2}$

$e^{2 x} v = 2 x e^{2 x} - e^{2 x} + C_{2}$

Step 8

Divide each term in $e^{2 x} v = 2 x e^{2 x} - e^{2 x} + C_{2}$ by $e^{2 x}$ and simplify.

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Step 8.1

Divide each term in $e^{2 x} v = 2 x e^{2 x} - e^{2 x} + C_{2}$ by $e^{2 x}$ .

$\frac{e^{2 x} v}{e^{2 x}} = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C_{2}}{e^{2 x}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{e^{2 x} v}{e^{2 x}} = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C_{2}}{e^{2 x}}$

Step 8.2.1.2

Divide $v$ by $1$ .

$v = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C_{2}}{e^{2 x}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Cancel the common factor.

$v = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C_{2}}{e^{2 x}}$

Step 8.3.1.1.2

Divide $2 x$ by $1$ .

$v = 2 x + \frac{- e^{2 x}}{e^{2 x}} + \frac{C_{2}}{e^{2 x}}$

Step 8.3.1.2

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

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Step 8.3.1.2.1

Cancel the common factor.

$v = 2 x + \frac{- e^{2 x}}{e^{2 x}} + \frac{C_{2}}{e^{2 x}}$

Step 8.3.1.2.2

Divide $- 1$ by $1$ .

$v = 2 x - 1 + \frac{C_{2}}{e^{2 x}}$

Step 9

Replace all occurrences of $v$ with $\frac{d y}{d x}$ .

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

Step 10

Rewrite the equation.

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

Step 11

Integrate both sides.

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Step 11.1

Set up an integral on each side.

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

Step 11.2

Apply the constant rule.

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

Step 11.3

Integrate the right side.

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Step 11.3.1

Split the single integral into multiple integrals.

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

Step 11.3.2

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

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

Step 11.3.3

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

$y + C_{3} = 2 (\frac{1}{2} x^{2} + C_{4}) + \int - 1 d x + \int \frac{C_{2}}{e^{2 x}} d x$

Step 11.3.4

Apply the constant rule.

$y + C_{3} = 2 (\frac{1}{2} x^{2} + C_{4}) - x + C_{5} + \int \frac{C_{2}}{e^{2 x}} d x$

Step 11.3.5

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

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + \int \frac{C_{2}}{e^{2 x}} d x$

Step 11.3.6

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

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int \frac{1}{e^{2 x}} d x$

Step 11.3.7

Simplify the expression.

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Step 11.3.7.1

Negate the exponent of $e^{2 x}$ and move it out of the denominator.

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int 1 {(e^{2 x})}^{- 1} d x$

Step 11.3.7.2

Simplify.

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Step 11.3.7.2.1

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

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Step 11.3.7.2.1.1

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

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int 1 e^{2 x \cdot - 1} d x$

Step 11.3.7.2.1.2

Multiply $- 1$ by $2$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int 1 e^{- 2 x} d x$

Step 11.3.7.2.2

Multiply $e^{- 2 x}$ by $1$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int e^{- 2 x} d x$

Step 11.3.8

Let $u_{2} = - 2 x$ . Then $d u_{2} = - 2 d x$ , so $- \frac{1}{2} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 11.3.8.1

Let $u_{2} = - 2 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 11.3.8.1.1

Differentiate $- 2 x$ .

$\frac{d}{d x} [- 2 x]$

Step 11.3.8.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$- 2 \frac{d}{d x} [x]$

Step 11.3.8.1.3

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

$- 2 \cdot 1$

Step 11.3.8.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 11.3.8.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int e^{u_{2}} \frac{1}{- 2} d u_{2}$

Step 11.3.9

Simplify.

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Step 11.3.9.1

Move the negative in front of the fraction.

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int e^{u_{2}} (- \frac{1}{2}) d u_{2}$

Step 11.3.9.2

Combine $e^{u_{2}}$ and $\frac{1}{2}$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} \int - \frac{e^{u_{2}}}{2} d u_{2}$

Step 11.3.10

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} (- \int \frac{e^{u_{2}}}{2} d u_{2})$

Step 11.3.11

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} (- (\frac{1}{2} \int e^{u_{2}} d u_{2}))$

Step 11.3.12

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) - x + C_{5} + C_{2} (- \frac{1}{2} (e^{u_{2}} + C_{6}))$

Step 11.3.13

Simplify.

$y + C_{3} = x^{2} - x + C_{2} (- \frac{1}{2} e^{u_{2}}) + C_{7}$

Step 11.3.14

Replace all occurrences of $u_{2}$ with $- 2 x$ .

$y + C_{3} = x^{2} - x + C_{2} (- \frac{1}{2} e^{- 2 x}) + C_{7}$

Step 11.3.15

Reorder terms.

$y + C_{3} = x^{2} - \frac{1}{2} e^{- 2 x} C_{2} - x + C_{7}$

Step 11.4

Group the constant of integration on the right side as $D$ .

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