Calculus Examples

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

Step 1

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

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

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 $e^{2 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

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

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

Step 7.4

Simplify.

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Step 7.4.1

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

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

Step 7.4.2

Cancel the common factor of $2$ .

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Step 7.4.2.1

Cancel the common factor.

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

Step 7.4.2.2

Rewrite the expression.

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

Step 7.4.3

Multiply $\int e^{2 x} (x) d x$ by $1$ .

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

Step 7.5

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

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

Step 7.6

Simplify.

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.7

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

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

Step 7.8

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

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Step 7.8.1

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

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Step 7.8.1.1

Differentiate $2 x$ .

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

Step 7.8.1.2

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 7.8.1.3

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

$2 \cdot 1$

Step 7.8.1.4

Multiply $2$ by $1$ .

$2$

Step 7.8.2

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

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

Step 7.9

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

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

Step 7.10

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

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

Step 7.11

Simplify.

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Step 7.11.1

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

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

Step 7.11.2

Multiply $2$ by $2$ .

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

Step 7.12

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

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

Step 7.13

Simplify.

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Step 7.13.1

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

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

Step 7.13.2

Simplify.

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Step 7.13.2.1

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

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

Step 7.13.2.2

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

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

Step 7.13.2.3

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

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

Step 7.13.2.4

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

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

Step 7.13.2.5

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

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

Step 7.13.2.6

To write $- (\frac{x e^{2 x}}{2} - \frac{e^{u}}{4})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.13.2.7

Combine $- (\frac{x e^{2 x}}{2} - \frac{e^{u}}{4})$ and $\frac{2}{2}$ .

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

Step 7.13.2.8

Combine the numerators over the common denominator.

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

Step 7.13.2.9

Multiply $2$ by $- 1$ .

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

Step 7.14

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

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

Step 7.15

Simplify.

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Step 7.15.1

Apply the distributive property.

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

Step 7.15.2

Cancel the common factor of $2$ .

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Step 7.15.2.1

Factor $2$ out of $- 2$ .

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

Step 7.15.2.2

Cancel the common factor.

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

Step 7.15.2.3

Rewrite the expression.

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

Step 7.15.3

Cancel the common factor of $2$ .

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Step 7.15.3.1

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

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

Step 7.15.3.2

Factor $2$ out of $- 2$ .

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

Step 7.15.3.3

Factor $2$ out of $4$ .

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

Step 7.15.3.4

Cancel the common factor.

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

Step 7.15.3.5

Rewrite the expression.

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

Step 7.15.4

Simplify each term.

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Step 7.15.4.1

Move the negative in front of the fraction.

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

Step 7.15.4.2

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

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Step 7.15.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 7.15.4.2.2

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

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

Step 7.15.5

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

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

Step 7.15.6

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

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

Step 7.15.7

Combine the numerators over the common denominator.

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

Step 7.15.8

Simplify the numerator.

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Step 7.15.8.1

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

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Step 7.15.8.1.1

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

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

Step 7.15.8.1.2

Multiply by $1$ .

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

Step 7.15.8.1.3

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

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

Step 7.15.8.2

Multiply $2$ by $- 1$ .

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

Step 7.15.9

Factor $- 1$ out of $- 2 x$ .

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

Step 7.15.10

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 7.15.11

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

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

Step 7.15.12

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

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

Step 7.15.13

Move the negative in front of the fraction.

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

Step 7.16

Reorder terms.

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

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Apply the distributive property.

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

Step 8.1.1.2

Rewrite using the commutative property of multiplication.

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

Step 8.1.1.3

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

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

Step 8.1.1.4

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

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

Step 8.1.1.5

Apply the distributive property.

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

Step 8.1.1.6

Cancel the common factor of $2$ .

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Step 8.1.1.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 8.1.1.6.2

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

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

Step 8.1.1.6.3

Cancel the common factor.

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

Step 8.1.1.6.4

Rewrite the expression.

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

Step 8.1.1.7

Multiply $- \frac{1}{2} (- e^{2 x})$ .

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Step 8.1.1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 8.1.1.7.2

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

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

Step 8.1.1.7.3

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

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

Step 8.1.1.8

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

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

Step 8.1.1.9

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

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Step 8.1.1.9.1

Move $e^{2 x}$ .

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

Step 8.1.1.9.2

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

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

Step 8.1.1.9.3

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

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

Step 8.1.1.9.4

Combine the numerators over the common denominator.

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

Step 8.1.1.10

Simplify the numerator.

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Step 8.1.1.10.1

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

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Step 8.1.1.10.1.1

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

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

Step 8.1.1.10.1.2

Multiply by $1$ .

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

Step 8.1.1.10.1.3

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

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

Step 8.1.1.10.2

Multiply $2$ by $- 1$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

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

Step 8.1.4

Combine the numerators over the common denominator.

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

Step 8.1.5

Simplify the numerator.

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Step 8.1.5.1

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

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Step 8.1.5.1.1

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

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

Step 8.1.5.1.2

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

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

Step 8.1.5.2

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

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

Step 8.1.6

Combine.

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

Step 8.1.7

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

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

Step 8.1.8

Multiply $2$ by $2$ .

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Combine the numerators over the common denominator.

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

Step 8.2.3.2

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

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

Step 8.2.3.3

Simplify terms.

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Step 8.2.3.3.1

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

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

Step 8.2.3.3.2

Combine the numerators over the common denominator.

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

Step 8.2.3.4

Simplify the numerator.

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Step 8.2.3.4.1

Apply the distributive property.

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

Step 8.2.3.4.2

Simplify.

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Step 8.2.3.4.2.1

Rewrite using the commutative property of multiplication.

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

Step 8.2.3.4.2.2

Rewrite using the commutative property of multiplication.

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

Step 8.2.3.4.2.3

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

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

Step 8.2.3.4.3

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

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

Step 8.2.3.5

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

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

Step 8.2.3.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.3.7

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

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