Calculus Examples

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

Step 1

Add $2 y$ to both sides of the equation.

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

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.

$e^{2 x + C}$

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 $e^{2 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.4

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

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

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 $d u_{1}$ .

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

Simplify.

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Step 7.8.1

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

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

Step 7.8.2

Multiply $2$ by $2$ .

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

Step 7.9

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

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

Step 7.10

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 7.10.1

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

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Step 7.10.1.1

Differentiate $2 x$ .

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

Step 7.10.1.2

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 7.10.1.3

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 7.10.1.4

Multiply $2$ by $1$ .

$2$

Step 7.10.2

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

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

Step 7.11

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

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

Step 7.12

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

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

Step 7.13

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

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

Step 7.14

Simplify.

$e^{2 x} y = \frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u_{1}} + \frac{1}{2} e^{u_{2}} + C$

Step 7.15

Substitute back in for each integration substitution variable.

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Step 7.15.1

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

$e^{2 x} y = \frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + \frac{1}{2} e^{u_{2}} + C$

Step 7.15.2

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

$e^{2 x} y = \frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + \frac{1}{2} e^{2 x} + C$

Step 8

Divide each term in $e^{2 x} y = \frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + \frac{1}{2} e^{2 x} + C$ by $e^{2 x}$ and simplify.

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Step 8.1

Divide each term in $e^{2 x} y = \frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + \frac{1}{2} e^{2 x} + C$ by $e^{2 x}$ .

$\frac{e^{2 x} y}{e^{2 x}} = \frac{\frac{1}{2} x e^{2 x}}{e^{2 x}} + \frac{- \frac{1}{4} e^{2 x}}{e^{2 x}} + \frac{\frac{1}{2} e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

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.

$\frac{e^{2 x} y}{e^{2 x}} = \frac{\frac{1}{2} x e^{2 x}}{e^{2 x}} + \frac{- \frac{1}{4} e^{2 x}}{e^{2 x}} + \frac{\frac{1}{2} e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{2} x e^{2 x}}{e^{2 x}} + \frac{- \frac{1}{4} e^{2 x}}{e^{2 x}} + \frac{\frac{1}{2} e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

$y = \frac{\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + \frac{1}{2} e^{2 x} + C}{e^{2 x}}$

Step 8.3.2

Simplify each term.

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Step 8.3.2.1

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

$y = \frac{\frac{x}{2} e^{2 x} - \frac{1}{4} e^{2 x} + \frac{1}{2} e^{2 x} + C}{e^{2 x}}$

Step 8.3.2.2

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

$y = \frac{\frac{x e^{2 x}}{2} - \frac{1}{4} e^{2 x} + \frac{1}{2} e^{2 x} + C}{e^{2 x}}$

Step 8.3.2.3

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

$y = \frac{\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + \frac{1}{2} e^{2 x} + C}{e^{2 x}}$

Step 8.3.2.4

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

$y = \frac{\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + \frac{e^{2 x}}{2} + C}{e^{2 x}}$

Step 8.3.3

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

$y = \frac{\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + \frac{e^{2 x}}{2} \cdot \frac{2}{2} + C}{e^{2 x}}$

Step 8.3.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.4.1

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

$y = \frac{\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + \frac{e^{2 x} \cdot 2}{2 \cdot 2} + C}{e^{2 x}}$

Step 8.3.4.2

Multiply $2$ by $2$ .

$y = \frac{\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + \frac{e^{2 x} \cdot 2}{4} + C}{e^{2 x}}$

Step 8.3.5

Combine the numerators over the common denominator.

$y = \frac{\frac{x e^{2 x}}{2} + \frac{- e^{2 x} + e^{2 x} \cdot 2}{4} + C}{e^{2 x}}$

Step 8.3.6

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

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Step 8.3.6.1

Reorder $e^{2 x}$ and $2$ .

$y = \frac{\frac{x e^{2 x}}{2} + \frac{- e^{2 x} + 2 \cdot e^{2 x}}{4} + C}{e^{2 x}}$

Step 8.3.6.2

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

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

Step 8.3.7

Simplify the numerator.

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Step 8.3.7.1

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

$y = \frac{\frac{x e^{2 x}}{2} \cdot \frac{2}{2} + \frac{e^{2 x}}{4} + C}{e^{2 x}}$

Step 8.3.7.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.7.2.1

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

$y = \frac{\frac{x e^{2 x} \cdot 2}{2 \cdot 2} + \frac{e^{2 x}}{4} + C}{e^{2 x}}$

Step 8.3.7.2.2

Multiply $2$ by $2$ .

$y = \frac{\frac{x e^{2 x} \cdot 2}{4} + \frac{e^{2 x}}{4} + C}{e^{2 x}}$

Step 8.3.7.3

Combine the numerators over the common denominator.

$y = \frac{\frac{x e^{2 x} \cdot 2 + e^{2 x}}{4} + C}{e^{2 x}}$

Step 8.3.7.4

Simplify the numerator.

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Step 8.3.7.4.1

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

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Step 8.3.7.4.1.1

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

$y = \frac{\frac{e^{2 x} (x \cdot 2) + e^{2 x}}{4} + C}{e^{2 x}}$

Step 8.3.7.4.1.2

Multiply by $1$ .

$y = \frac{\frac{e^{2 x} (x \cdot 2) + e^{2 x} \cdot 1}{4} + C}{e^{2 x}}$

Step 8.3.7.4.1.3

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

$y = \frac{\frac{e^{2 x} (x \cdot 2 + 1)}{4} + C}{e^{2 x}}$

Step 8.3.7.4.2

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

$y = \frac{\frac{e^{2 x} (2 x + 1)}{4} + C}{e^{2 x}}$

Step 8.3.7.5

To write $C$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{\frac{e^{2 x} (2 x + 1)}{4} + C \cdot \frac{4}{4}}{e^{2 x}}$

Step 8.3.7.6

Combine $C$ and $\frac{4}{4}$ .

$y = \frac{\frac{e^{2 x} (2 x + 1)}{4} + \frac{C \cdot 4}{4}}{e^{2 x}}$

Step 8.3.7.7

Combine the numerators over the common denominator.

$y = \frac{\frac{e^{2 x} (2 x + 1) + C \cdot 4}{4}}{e^{2 x}}$

Step 8.3.7.8

Simplify the numerator.

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Step 8.3.7.8.1

Apply the distributive property.

$y = \frac{\frac{e^{2 x} (2 x) + e^{2 x} \cdot 1 + C \cdot 4}{4}}{e^{2 x}}$

Step 8.3.7.8.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{2 e^{2 x} x + e^{2 x} \cdot 1 + C \cdot 4}{4}}{e^{2 x}}$

Step 8.3.7.8.3

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

$y = \frac{\frac{2 e^{2 x} x + e^{2 x} + C \cdot 4}{4}}{e^{2 x}}$

Step 8.3.7.8.4

Move $4$ to the left of $C$ .

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

Step 8.3.8

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{2 e^{2 x} x + e^{2 x} + 4 C}{4} \cdot \frac{1}{e^{2 x}}$

Step 8.3.9

Multiply $\frac{2 e^{2 x} x + e^{2 x} + 4 C}{4}$ by $\frac{1}{e^{2 x}}$ .

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

Step 8.3.10

Reorder factors in $\frac{2 e^{2 x} x + e^{2 x} + 4 C}{4 e^{2 x}}$ .

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