Calculus Examples

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

Step 1

Subtract $y$ from both sides of the equation.

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

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.

$e^{\int - 1 d x}$

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 $e^{- x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7.2

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

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

Step 7.3

Multiply $2$ by $- 1$ .

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

Step 7.4

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

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

Step 7.5

Multiply $- 2$ by $- 1$ .

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

Step 7.6

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

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

Step 7.7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7.8

Simplify.

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Step 7.8.1

Multiply $- 1$ by $- 1$ .

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

Step 7.8.2

Multiply $\int e^{- x} d x$ by $1$ .

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

Step 7.9

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.9.1

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

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Step 7.9.1.1

Differentiate $- x$ .

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

Step 7.9.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 7.9.1.3

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

$- 1 \cdot 1$

Step 7.9.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.9.2

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

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

Step 7.10

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 7.11

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

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

Step 7.12

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

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

Step 7.13

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

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

Step 7.14

Simplify.

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Step 7.14.1

Apply the distributive property.

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

Step 7.14.2

Simplify.

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Step 7.14.2.1

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

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Step 7.14.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 7.14.2.1.2

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

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

Step 7.14.2.2

Multiply $- 2$ by $- 1$ .

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

Step 7.14.2.3

Multiply $- 2$ by $- 1$ .

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

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

Step 8

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

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Step 8.1

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

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

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.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Divide $x^{2}$ by $1$ .

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

Step 8.3.1.2

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

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Step 8.3.1.2.1

Cancel the common factor.

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

Step 8.3.1.2.2

Divide $2 x$ by $1$ .

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

Step 8.3.1.3

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

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Step 8.3.1.3.1

Cancel the common factor.

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

Step 8.3.1.3.2

Divide $2$ by $1$ .

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

Step 9

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $0$ for $y$ in $y = x^{2} + 2 x + 2 + \frac{C}{e^{- x}}$ .

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

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $0^{2} + 2 \cdot 0 + 2 + \frac{C}{e^{0}} = 0$ .

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

Step 10.2

Simplify $0^{2} + 2 \cdot 0 + 2 + \frac{C}{e^{0}}$ .

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Raising $0$ to any positive power yields $0$ .

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

Step 10.2.1.2

Multiply $2$ by $0$ .

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

Step 10.2.1.3

Anything raised to $0$ is $1$ .

$0 + 0 + 2 + \frac{C}{1} = 0$

Step 10.2.1.4

Divide $C$ by $1$ .

$0 + 0 + 2 + C = 0$

Step 10.2.2

Combine the opposite terms in $0 + 0 + 2 + C$ .

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Step 10.2.2.1

Add $0$ and $0$ .

$0 + 2 + C = 0$

Step 10.2.2.2

Add $0$ and $2$ .

$2 + C = 0$

Step 10.3

Subtract $2$ from both sides of the equation.

$C = - 2$

Step 11

Substitute $- 2$ for $C$ in $y = x^{2} + 2 x + 2 + \frac{C}{e^{- x}}$ and simplify.

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Step 11.1

Substitute $- 2$ for $C$ .

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

Step 11.2

Move the negative in front of the fraction.

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