Calculus Examples

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

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 $x$ .

$\frac{d v}{d x} + v = \cos (x)$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int d x}$

Step 2.2

Apply the constant rule.

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

Step 2.3

Remove the constant of integration.

$e^{x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Reorder factors in $e^{x} \frac{d v}{d x} + e^{x} v = e^{x} \cos (x)$ .

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

Step 4

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

$\frac{d}{d x} [e^{x} v] = e^{x} \cos (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x} v] d x = \int e^{x} \cos (x) d x$

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Reorder $e^{x}$ and $\cos (x)$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Simplify the expression.

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.4.2

Multiply $\int e^{x} (\sin (x)) d x$ by $1$ .

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

Step 7.4.3

Reorder $e^{x}$ and $\sin (x)$ .

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

Step 7.5

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

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

Step 7.6

Solving for $\int e^{x} \cos (x) d x$ , we find that $\int e^{x} \cos (x) d x$ = $\frac{\cos (x) e^{x} + \sin (x) e^{x}}{2}$ .

$e^{x} v = \frac{\cos (x) e^{x} + \sin (x) e^{x}}{2} + C_{2}$

Step 7.7

Rewrite $\frac{\cos (x) e^{x} + \sin (x) e^{x}}{2} + C_{2}$ as $\frac{1}{2} (\cos (x) e^{x} + \sin (x) e^{x}) + C_{2}$ .

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

Step 8

Divide each term in $e^{x} v = \frac{1}{2} (\cos (x) e^{x} + \sin (x) e^{x}) + C_{2}$ by $e^{x}$ and simplify.

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Step 8.1

Divide each term in $e^{x} v = \frac{1}{2} (\cos (x) e^{x} + \sin (x) e^{x}) + C_{2}$ by $e^{x}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $v$ by $1$ .

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

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

Simplify the numerator.

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Step 8.3.1.1.1

Factor $e^{x}$ out of $\cos (x) e^{x} + \sin (x) e^{x}$ .

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Step 8.3.1.1.1.1

Factor $e^{x}$ out of $\cos (x) e^{x}$ .

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

Step 8.3.1.1.1.2

Factor $e^{x}$ out of $\sin (x) e^{x}$ .

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

Step 8.3.1.1.1.3

Factor $e^{x}$ out of $e^{x} \cos (x) + e^{x} \sin (x)$ .

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

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

Step 8.3.1.1.2

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

$v = \frac{\frac{e^{x}}{2} (\cos (x) + \sin (x))}{e^{x}} + \frac{C_{2}}{e^{x}}$

Step 8.3.1.2

Factor $\frac{e^{x}}{2}$ out of $\frac{\frac{e^{x}}{2} (\cos (x) + \sin (x))}{e^{x}}$ .

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

Step 8.3.1.3

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

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Step 8.3.1.3.1

Cancel the common factor.

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

Step 8.3.1.3.2

Rewrite the expression.

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

Step 8.3.1.4

Apply the distributive property.

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

Step 8.3.1.5

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

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

Step 8.3.1.6

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

$v = \frac{\cos (x)}{2} + \frac{\sin (x)}{2} + \frac{C_{2}}{e^{x}}$

Step 9

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

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

Step 10

Rewrite the equation.

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

Step 11

Integrate both sides.

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Step 11.1

Set up an integral on each side.

$\int d y = \int \frac{\cos (x)}{2} + \frac{\sin (x)}{2} + \frac{C_{2}}{e^{x}} d x$

Step 11.2

Apply the constant rule.

$y + C_{3} = \int \frac{\cos (x)}{2} + \frac{\sin (x)}{2} + \frac{C_{2}}{e^{x}} d x$

Step 11.3

Integrate the right side.

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Step 11.3.1

Split the single integral into multiple integrals.

$y + C_{3} = \int \frac{\cos (x)}{2} d x + \int \frac{\sin (x)}{2} d x + \int \frac{C_{2}}{e^{x}} d x$

Step 11.3.2

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

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

Step 11.3.3

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

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

Step 11.3.4

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

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

Step 11.3.5

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

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

Step 11.3.6

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

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

Step 11.3.7

Simplify the expression.

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Step 11.3.7.1

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

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

Step 11.3.7.2

Simplify.

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Step 11.3.7.2.1

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

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Step 11.3.7.2.1.1

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

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

Step 11.3.7.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 11.3.7.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 11.3.7.2.2

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

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

Step 11.3.8

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 11.3.8.1

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

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Step 11.3.8.1.1

Differentiate $- x$ .

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

Step 11.3.8.1.2

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 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 $n = 1$ .

$- 1 \cdot 1$

Step 11.3.8.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 11.3.8.2

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

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

Step 11.3.9

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

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

Step 11.3.10

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

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

Step 11.3.11

Simplify.

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

Step 11.3.12

Replace all occurrences of $u$ with $- x$ .

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

Step 11.3.13

Reorder terms.

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

Step 11.3.14

Reorder terms.

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

Step 11.4

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

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