Calculus Examples

$\frac{d y}{d x} + y - x^{2} - 2 x = 0$

Step 1

Rewrite the equation as $M (x) \frac{d y}{d x} + P (x) y = Q (x)$ .

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Step 1.1

Add $x^{2}$ to both sides of the equation.

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

Step 1.2

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

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 1$ .

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Step 2.1

Set up the integration.

$e^{\int d x}$

Step 2.2

Apply the constant rule.

$e^{x + C}$

Step 2.3

Remove the constant of integration.

$e^{x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Multiply $2$ by $- 1$ .

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

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

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

Step 7.9

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

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

Step 7.10

Simplify.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

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

Step 8.3.2

Simplify each term.

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Step 8.3.2.1

Apply the distributive property.

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

Step 8.3.2.2

Multiply $- 1$ by $2$ .

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

Step 8.3.3

Combine the opposite terms in $x^{2} e^{x} - 2 x e^{x} + 2 e^{x} + 2 x e^{x} - 2 e^{x} + C$ .

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Step 8.3.3.1

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

$y = \frac{x^{2} e^{x} + 2 e^{x} + 0 - 2 e^{x} + C}{e^{x}}$

Step 8.3.3.2

Add $x^{2} e^{x} + 2 e^{x}$ and $0$ .

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

Step 8.3.3.3

Subtract $2 e^{x}$ from $2 e^{x}$ .

$y = \frac{x^{2} e^{x} + 0 + C}{e^{x}}$

Step 8.3.3.4

Add $x^{2} e^{x}$ and $0$ .

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