Calculus Examples

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

Step 1

Add $3 \cdot y$ to both sides of the equation.

$\frac{d y}{d x} + 3 \cdot y = 2 \cdot x + 1$

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.

$e^{3 x + C}$

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 $e^{3 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

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

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

Step 3.4

Reorder factors in $e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 2 e^{3 x} x + e^{3 x}$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{3 x}$ .

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

Step 7.4

Simplify.

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Step 7.4.1

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

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

Step 7.4.2

Combine $x$ and $\frac{e^{3 x}}{3}$ .

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

Step 7.4.3

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

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

Step 7.5

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

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

Step 7.6

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 7.6.1

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

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Step 7.6.1.1

Differentiate $3 x$ .

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

Step 7.6.1.2

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 7.6.1.3

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 7.6.1.4

Multiply $3$ by $1$ .

$3$

Step 7.6.2

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

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

Step 7.7

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

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

Step 7.8

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

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

Step 7.9

Simplify.

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Step 7.9.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 7.9.2

Multiply $3$ by $3$ .

$e^{3 x} y = 2 (\frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u_{1}} d u_{1}) + \int e^{3 x} d x$

Step 7.10

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

$e^{3 x} y = 2 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int e^{3 x} d x$

Step 7.11

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 7.11.1

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

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Step 7.11.1.1

Differentiate $3 x$ .

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

Step 7.11.1.2

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 7.11.1.3

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 7.11.1.4

Multiply $3$ by $1$ .

$3$

Step 7.11.2

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

$e^{3 x} y = 2 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int e^{u_{2}} \frac{1}{3} d u_{2}$

Step 7.12

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

$e^{3 x} y = 2 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int \frac{e^{u_{2}}}{3} d u_{2}$

Step 7.13

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

$e^{3 x} y = 2 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \frac{1}{3} \int e^{u_{2}} d u_{2}$

Step 7.14

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

$e^{3 x} y = 2 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \frac{1}{3} (e^{u_{2}} + C)$

Step 7.15

Simplify.

$e^{3 x} y = 2 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u_{1}}) + \frac{1}{3} e^{u_{2}} + C$

Step 7.16

Substitute back in for each integration substitution variable.

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Step 7.16.1

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

$e^{3 x} y = 2 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x}) + \frac{1}{3} e^{u_{2}} + C$

Step 7.16.2

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

$e^{3 x} y = 2 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x}) + \frac{1}{3} e^{3 x} + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

Combine $e^{3 x}$ and $\frac{1}{9}$ .

$e^{3 x} y = 2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C$

Step 8.1.2

Remove parentheses.

$e^{3 x} y = 2 (\frac{1}{3} \cdot x e^{3 x} - \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C$

Step 8.2

Divide each term in $e^{3 x} y = 2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C$ by $e^{3 x}$ and simplify.

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Step 8.2.1

Divide each term in $e^{3 x} y = 2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C$ by $e^{3 x}$ .

$\frac{e^{3 x} y}{e^{3 x}} = \frac{2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9})}{e^{3 x}} + \frac{\frac{1}{3} \cdot e^{3 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{e^{3 x} y}{e^{3 x}} = \frac{2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9})}{e^{3 x}} + \frac{\frac{1}{3} \cdot e^{3 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9})}{e^{3 x}} + \frac{\frac{1}{3} \cdot e^{3 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Combine the numerators over the common denominator.

$y = \frac{2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2

Simplify each term.

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Step 8.2.3.2.1

Multiply $\frac{1}{3} (x e^{3 x})$ .

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Step 8.2.3.2.1.1

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

$y = \frac{2 (\frac{x}{3} e^{3 x} - \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2.1.2

Combine $\frac{x}{3}$ and $e^{3 x}$ .

$y = \frac{2 (\frac{x e^{3 x}}{3} - \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2.2

Apply the distributive property.

$y = \frac{2 \frac{x e^{3 x}}{3} + 2 (- \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2.3

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

$y = \frac{\frac{2 (x e^{3 x})}{3} + 2 (- \frac{e^{3 x}}{9}) + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2.4

Multiply $2 (- \frac{e^{3 x}}{9})$ .

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Step 8.2.3.2.4.1

Multiply $- 1$ by $2$ .

$y = \frac{\frac{2 (x e^{3 x})}{3} - 2 \frac{e^{3 x}}{9} + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2.4.2

Combine $- 2$ and $\frac{e^{3 x}}{9}$ .

$y = \frac{\frac{2 (x e^{3 x})}{3} + \frac{- 2 e^{3 x}}{9} + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2.5

Move the negative in front of the fraction.

$y = \frac{\frac{2 x e^{3 x}}{3} - \frac{2 e^{3 x}}{9} + \frac{1}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 8.2.3.2.6

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

$y = \frac{\frac{2 x e^{3 x}}{3} - \frac{2 e^{3 x}}{9} + \frac{e^{3 x}}{3} + C}{e^{3 x}}$

Step 8.2.3.3

To write $\frac{e^{3 x}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{\frac{2 x e^{3 x}}{3} - \frac{2 e^{3 x}}{9} + \frac{e^{3 x}}{3} \cdot \frac{3}{3} + C}{e^{3 x}}$

Step 8.2.3.4

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.4.1

Multiply $\frac{e^{3 x}}{3}$ by $\frac{3}{3}$ .

$y = \frac{\frac{2 x e^{3 x}}{3} - \frac{2 e^{3 x}}{9} + \frac{e^{3 x} \cdot 3}{3 \cdot 3} + C}{e^{3 x}}$

Step 8.2.3.4.2

Multiply $3$ by $3$ .

$y = \frac{\frac{2 x e^{3 x}}{3} - \frac{2 e^{3 x}}{9} + \frac{e^{3 x} \cdot 3}{9} + C}{e^{3 x}}$

Step 8.2.3.5

Combine the numerators over the common denominator.

$y = \frac{\frac{2 x e^{3 x}}{3} + \frac{- 2 e^{3 x} + e^{3 x} \cdot 3}{9} + C}{e^{3 x}}$

Step 8.2.3.6

Add $- 2 e^{3 x}$ and $e^{3 x} \cdot 3$ .

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Step 8.2.3.6.1

Reorder $e^{3 x}$ and $3$ .

$y = \frac{\frac{2 x e^{3 x}}{3} + \frac{- 2 e^{3 x} + 3 \cdot e^{3 x}}{9} + C}{e^{3 x}}$

Step 8.2.3.6.2

Add $- 2 e^{3 x}$ and $3 \cdot e^{3 x}$ .

$y = \frac{\frac{2 x e^{3 x}}{3} + \frac{e^{3 x}}{9} + C}{e^{3 x}}$

Step 8.2.3.7

Simplify the numerator.

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Step 8.2.3.7.1

To write $\frac{2 x e^{3 x}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{\frac{2 x e^{3 x}}{3} \cdot \frac{3}{3} + \frac{e^{3 x}}{9} + C}{e^{3 x}}$

Step 8.2.3.7.2

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.7.2.1

Multiply $\frac{2 x e^{3 x}}{3}$ by $\frac{3}{3}$ .

$y = \frac{\frac{2 x e^{3 x} \cdot 3}{3 \cdot 3} + \frac{e^{3 x}}{9} + C}{e^{3 x}}$

Step 8.2.3.7.2.2

Multiply $3$ by $3$ .

$y = \frac{\frac{2 x e^{3 x} \cdot 3}{9} + \frac{e^{3 x}}{9} + C}{e^{3 x}}$

Step 8.2.3.7.3

Combine the numerators over the common denominator.

$y = \frac{\frac{2 x e^{3 x} \cdot 3 + e^{3 x}}{9} + C}{e^{3 x}}$

Step 8.2.3.7.4

Simplify the numerator.

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Step 8.2.3.7.4.1

Factor $e^{3 x}$ out of $2 x e^{3 x} \cdot 3 + e^{3 x}$ .

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Step 8.2.3.7.4.1.1

Factor $e^{3 x}$ out of $2 x e^{3 x} \cdot 3$ .

$y = \frac{\frac{e^{3 x} (2 x \cdot 3) + e^{3 x}}{9} + C}{e^{3 x}}$

Step 8.2.3.7.4.1.2

Multiply by $1$ .

$y = \frac{\frac{e^{3 x} (2 x \cdot 3) + e^{3 x} \cdot 1}{9} + C}{e^{3 x}}$

Step 8.2.3.7.4.1.3

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

$y = \frac{\frac{e^{3 x} (2 x \cdot 3 + 1)}{9} + C}{e^{3 x}}$

Step 8.2.3.7.4.2

Multiply $3$ by $2$ .

$y = \frac{\frac{e^{3 x} (6 x + 1)}{9} + C}{e^{3 x}}$

Step 8.2.3.7.5

To write $C$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \frac{\frac{e^{3 x} (6 x + 1)}{9} + C \cdot \frac{9}{9}}{e^{3 x}}$

Step 8.2.3.7.6

Combine $C$ and $\frac{9}{9}$ .

$y = \frac{\frac{e^{3 x} (6 x + 1)}{9} + \frac{C \cdot 9}{9}}{e^{3 x}}$

Step 8.2.3.7.7

Combine the numerators over the common denominator.

$y = \frac{\frac{e^{3 x} (6 x + 1) + C \cdot 9}{9}}{e^{3 x}}$

Step 8.2.3.7.8

Simplify the numerator.

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Step 8.2.3.7.8.1

Apply the distributive property.

$y = \frac{\frac{e^{3 x} (6 x) + e^{3 x} \cdot 1 + C \cdot 9}{9}}{e^{3 x}}$

Step 8.2.3.7.8.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{6 e^{3 x} x + e^{3 x} \cdot 1 + C \cdot 9}{9}}{e^{3 x}}$

Step 8.2.3.7.8.3

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

$y = \frac{\frac{6 e^{3 x} x + e^{3 x} + C \cdot 9}{9}}{e^{3 x}}$

Step 8.2.3.7.8.4

Move $9$ to the left of $C$ .

$y = \frac{\frac{6 e^{3 x} x + e^{3 x} + 9 C}{9}}{e^{3 x}}$

Step 8.2.3.8

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{6 e^{3 x} x + e^{3 x} + 9 C}{9} \cdot \frac{1}{e^{3 x}}$

Step 8.2.3.9

Multiply $\frac{6 e^{3 x} x + e^{3 x} + 9 C}{9}$ by $\frac{1}{e^{3 x}}$ .

$y = \frac{6 e^{3 x} x + e^{3 x} + 9 C}{9 e^{3 x}}$

Step 8.2.3.10

Reorder factors in $\frac{6 e^{3 x} x + e^{3 x} + 9 C}{9 e^{3 x}}$ .

$y = \frac{6 x e^{3 x} + e^{3 x} + 9 C}{9 e^{3 x}}$