Calculus Examples

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

Step 1

Subtract $y$ from both sides of the equation.

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

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 $e^{- x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Simplify each term.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

Multiply $2$ by $- 1$ .

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

Step 7.4

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

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

Step 7.5

Multiply $- 2$ by $- 1$ .

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

Simplify.

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Step 7.8.1

Multiply $- 1$ by $- 1$ .

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

Step 7.8.2

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

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

Step 7.9

Let $u_{1} = - x$ . Then $d u_{1} = - d x$ , so $- d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.9.1

Let $u_{1} = - x$ . Find $\frac{d u_{1}}{d x}$ .

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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 $u_{1}$ and $d u_{1}$ .

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

Step 7.10

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

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

Step 7.11

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

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

Step 7.12

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

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

Step 7.13

Let $u_{2} = - x$ . Then $d u_{2} = - d x$ , so $- d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.13.1

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

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Step 7.13.1.1

Differentiate $- x$ .

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

Step 7.13.1.2

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 7.13.1.3

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 7.13.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.13.2

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

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

Step 7.14

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

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

Step 7.15

Simplify.

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Step 7.15.1

Multiply $- 1$ by $- 1$ .

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

Step 7.15.2

Multiply $\int e^{u_{2}} d u_{2}$ by $1$ .

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

Step 7.16

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

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

Step 7.17

Simplify.

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Step 7.17.1

Simplify.

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

Step 7.17.2

Add $- 2 e^{u_{1}}$ and $e^{u_{2}}$ .

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

Step 7.18

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

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

Step 8

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

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Step 8.1

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

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

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.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 8.3.1.1.2

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

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

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.

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

Step 8.3.1.2.2

Divide $- 2 x$ by $1$ .

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

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.

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

Step 8.3.1.3.2

Divide $- 1$ by $1$ .

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