Calculus Examples

$\frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} = 3$

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 $x$ .

$\frac{d v}{d x} + 3 v = 3$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 3$ .

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Step 2.1

Set up the integration.

$e^{\int 3 d x}$

Step 2.2

Apply the constant rule.

$e^{3 x + C_{1}}$

Step 2.3

Remove the constant of integration.

$e^{3 x}$

Step 3

Multiply each term by the integrating factor $e^{3 x}$ .

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Step 3.1

Multiply each term by $e^{3 x}$ .

$e^{3 x} \frac{d v}{d x} + e^{3 x} (3 v) = e^{3 x} \cdot 3$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{3 x} \frac{d v}{d x} + 3 e^{3 x} v = e^{3 x} \cdot 3$

Step 3.3

Move $3$ to the left of $e^{3 x}$ .

$e^{3 x} \frac{d v}{d x} + 3 e^{3 x} v = 3 \cdot e^{3 x}$

Step 3.4

Reorder factors in $e^{3 x} \frac{d v}{d x} + 3 e^{3 x} v = 3 e^{3 x}$ .

$e^{3 x} \frac{d v}{d x} + 3 v e^{3 x} = 3 e^{3 x}$

Step 4

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

$\frac{d}{d x} [e^{3 x} v] = 3 e^{3 x}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{3 x} v] d x = \int 3 e^{3 x} d x$

Step 6

Integrate the left side.

$e^{3 x} v = \int 3 e^{3 x} d x$

Step 7

Integrate the right side.

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Step 7.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{3 x} v = 3 \int e^{3 x} d x$

Step 7.2

Let $u_{1} = 3 x$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.2.1

Let $u_{1} = 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 7.2.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 7.2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 7.2.1.3

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

$3 \cdot 1$

Step 7.2.1.4

Multiply $3$ by $1$ .

$3$

Step 7.2.2

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

$e^{3 x} v = 3 \int e^{u_{1}} \frac{1}{3} d u_{1}$

Step 7.3

Combine $e^{u_{1}}$ and $\frac{1}{3}$ .

$e^{3 x} v = 3 \int \frac{e^{u_{1}}}{3} d u_{1}$

Step 7.4

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$e^{3 x} v = 3 (\frac{1}{3} \int e^{u_{1}} d u_{1})$

Step 7.5

Simplify.

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Step 7.5.1

Combine $\frac{1}{3}$ and $3$ .

$e^{3 x} v = \frac{3}{3} \int e^{u_{1}} d u_{1}$

Step 7.5.2

Cancel the common factor of $3$ .

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Step 7.5.2.1

Cancel the common factor.

$e^{3 x} v = \frac{3}{3} \int e^{u_{1}} d u_{1}$

Step 7.5.2.2

Rewrite the expression.

$e^{3 x} v = 1 \int e^{u_{1}} d u_{1}$

Step 7.5.3

Multiply $\int e^{u_{1}} d u_{1}$ by $1$ .

$e^{3 x} v = \int e^{u_{1}} d u_{1}$

Step 7.6

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

$e^{3 x} v = e^{u_{1}} + C_{2}$

Step 7.7

Replace all occurrences of $u_{1}$ with $3 x$ .

$e^{3 x} v = e^{3 x} + C_{2}$

Step 8

Divide each term in $e^{3 x} v = e^{3 x} + C_{2}$ by $e^{3 x}$ and simplify.

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Step 8.1

Divide each term in $e^{3 x} v = e^{3 x} + C_{2}$ by $e^{3 x}$ .

$\frac{e^{3 x} v}{e^{3 x}} = \frac{e^{3 x}}{e^{3 x}} + \frac{C_{2}}{e^{3 x}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $e^{3 x}$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{e^{3 x} v}{e^{3 x}} = \frac{e^{3 x}}{e^{3 x}} + \frac{C_{2}}{e^{3 x}}$

Step 8.2.1.2

Divide $v$ by $1$ .

$v = \frac{e^{3 x}}{e^{3 x}} + \frac{C_{2}}{e^{3 x}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $e^{3 x}$ .

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Step 8.3.1.1

Cancel the common factor.

$v = \frac{e^{3 x}}{e^{3 x}} + \frac{C_{2}}{e^{3 x}}$

Step 8.3.1.2

Rewrite the expression.

$v = 1 + \frac{C_{2}}{e^{3 x}}$

Step 9

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

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

Step 10

Rewrite the equation.

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

Step 11

Integrate both sides.

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Step 11.1

Set up an integral on each side.

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

Step 11.2

Apply the constant rule.

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

Step 11.3

Integrate the right side.

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Step 11.3.1

Split the single integral into multiple integrals.

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

Step 11.3.2

Apply the constant rule.

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

Step 11.3.3

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

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

Step 11.3.4

Simplify the expression.

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Step 11.3.4.1

Negate the exponent of $e^{3 x}$ and move it out of the denominator.

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

Step 11.3.4.2

Simplify.

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Step 11.3.4.2.1

Multiply the exponents in ${(e^{3 x})}^{- 1}$ .

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Step 11.3.4.2.1.1

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

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

Step 11.3.4.2.1.2

Multiply $- 1$ by $3$ .

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

Step 11.3.4.2.2

Multiply $e^{- 3 x}$ by $1$ .

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

Step 11.3.5

Let $u_{2} = - 3 x$ . Then $d u_{2} = - 3 d x$ , so $- \frac{1}{3} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 11.3.5.1

Let $u_{2} = - 3 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 11.3.5.1.1

Differentiate $- 3 x$ .

$\frac{d}{d x} [- 3 x]$

Step 11.3.5.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 11.3.5.1.3

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

$- 3 \cdot 1$

Step 11.3.5.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 11.3.5.2

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

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

Step 11.3.6

Simplify.

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Step 11.3.6.1

Move the negative in front of the fraction.

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

Step 11.3.6.2

Combine $e^{u_{2}}$ and $\frac{1}{3}$ .

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

Step 11.3.7

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

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

Step 11.3.8

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

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

Step 11.3.9

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

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

Step 11.3.10

Simplify.

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Step 11.3.10.1

Simplify.

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

Step 11.3.10.2

Combine $e^{u_{2}}$ and $\frac{1}{3}$ .

$y + C_{3} = x - \frac{e^{u_{2}}}{3} C_{2} + C_{6}$

Step 11.3.11

Replace all occurrences of $u_{2}$ with $- 3 x$ .

$y + C_{3} = x - \frac{e^{- 3 x}}{3} C_{2} + C_{6}$

Step 11.3.12

Reorder terms.

$y + C_{3} = x - \frac{1}{3} e^{- 3 x} C_{2} + C_{6}$

Step 11.3.13

Reorder terms.

$y + C_{3} = - \frac{1}{3} e^{- 3 x} C_{2} + x + C_{6}$

Step 11.4

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

$y = - \frac{1}{3} e^{- 3 x} C + x + D$