Calculus Examples

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

Step 1

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

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

Step 2

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

Tap for more steps...

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 $e^{2 x}$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify.

Tap for more steps...

Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

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

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

Step 7.4

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

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

Step 7.5

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

Tap for more steps...

Step 7.5.1

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

Tap for more steps...

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 $u$ and $d u$ .

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

Simplify.

Tap for more steps...

Step 7.8.1

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

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

Step 7.8.2

Multiply $2$ by $2$ .

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

Step 7.9

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

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

Step 7.10

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

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

Step 7.11

Replace all occurrences of $u$ with $2 x$ .

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

Step 7.12

Simplify.

Tap for more steps...

Step 7.12.1

Simplify each term.

Tap for more steps...

Step 7.12.1.1

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

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

Step 7.12.1.2

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

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

Step 7.12.1.3

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

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

Step 7.12.2

Apply the distributive property.

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

Step 7.12.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.12.3.1

Cancel the common factor.

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

Step 7.12.3.2

Rewrite the expression.

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

Step 7.12.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.12.4.1

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

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

Step 7.12.4.2

Factor $2$ out of $4$ .

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

Step 7.12.4.3

Cancel the common factor.

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

Step 7.12.4.4

Rewrite the expression.

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

Step 7.12.5

Move the negative in front of the fraction.

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

Step 7.12.6

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

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

Step 7.12.7

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

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

Step 7.12.8

Combine the numerators over the common denominator.

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

Step 7.12.9

Simplify the numerator.

Tap for more steps...

Step 7.12.9.1

Move $2$ to the left of $x e^{2 x}$ .

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

Step 7.12.9.2

Factor $e^{2 x}$ out of $2 x e^{2 x} - e^{2 x}$ .

Tap for more steps...

Step 7.12.9.2.1

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

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

Step 7.12.9.2.2

Factor $e^{2 x}$ out of $- e^{2 x}$ .

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

Step 7.12.9.2.3

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

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

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

Step 8

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

Tap for more steps...

Step 8.1

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

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

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

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

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Combine the numerators over the common denominator.

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

Step 8.3.2

Simplify each term.

Tap for more steps...

Step 8.3.2.1

Apply the distributive property.

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

Step 8.3.2.2

Rewrite using the commutative property of multiplication.

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

Step 8.3.2.3

Move $- 1$ to the left of $e^{2 x}$ .

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

Step 8.3.2.4

Rewrite $- 1 e^{2 x}$ as $- e^{2 x}$ .

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

Step 8.3.2.5

Apply the distributive property.

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

Step 8.3.2.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.3.2.6.1

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

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

Step 8.3.2.6.2

Cancel the common factor.

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

Step 8.3.2.6.3

Rewrite the expression.

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

Step 8.3.2.7

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

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

Step 8.3.2.8

Combine $e^{2 x} x$ and $- \frac{e^{2 x}}{2}$ using a common denominator.

Tap for more steps...

Step 8.3.2.8.1

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

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

Step 8.3.2.8.2

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

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

Step 8.3.2.8.3

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

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

Step 8.3.2.8.4

Combine the numerators over the common denominator.

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

Step 8.3.2.9

Simplify the numerator.

Tap for more steps...

Step 8.3.2.9.1

Move $2$ to the left of $x e^{2 x}$ .

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

Step 8.3.2.9.2

Factor $e^{2 x}$ out of $2 x e^{2 x} - e^{2 x}$ .

Tap for more steps...

Step 8.3.2.9.2.1

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

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

Step 8.3.2.9.2.2

Factor $e^{2 x}$ out of $- e^{2 x}$ .

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

Step 8.3.2.9.2.3

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

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

Step 8.3.3

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

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

Step 8.3.4

Simplify terms.

Tap for more steps...

Step 8.3.4.1

Combine $C$ and $\frac{2}{2}$ .

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

Step 8.3.4.2

Combine the numerators over the common denominator.

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

Step 8.3.5

Simplify the numerator.

Tap for more steps...

Step 8.3.5.1

Apply the distributive property.

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

Step 8.3.5.2

Rewrite using the commutative property of multiplication.

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

Step 8.3.5.3

Move $- 1$ to the left of $e^{2 x}$ .

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

Step 8.3.5.4

Rewrite $- 1 e^{2 x}$ as $- e^{2 x}$ .

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

Step 8.3.5.5

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

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

Step 8.3.6

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

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

Step 8.3.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3.8

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

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