Calculus Examples

$\frac{d x}{d y} = 2 y + x + 3$

Step 1

Subtract $x$ from both sides of the equation.

$\frac{d x}{d y} - x = 2 y + 3$

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Apply the constant rule.

$e^{- y + C}$

Step 2.3

Remove the constant of integration.

$e^{- y}$

Step 3

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

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Step 3.1

Multiply each term by $e^{- y}$ .

$e^{- y} \frac{d x}{d y} + e^{- y} (- x) = e^{- y} (2 y) + e^{- y} \cdot 3$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- y} \frac{d x}{d y} - e^{- y} x = e^{- y} (2 y) + e^{- y} \cdot 3$

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

$e^{- y} \frac{d x}{d y} - e^{- y} x = 2 e^{- y} y + e^{- y} \cdot 3$

Step 3.3.2

Move $3$ to the left of $e^{- y}$ .

$e^{- y} \frac{d x}{d y} - e^{- y} x = 2 e^{- y} y + 3 e^{- y}$

Step 3.4

Reorder factors in $e^{- y} \frac{d x}{d y} - e^{- y} x = 2 e^{- y} y + 3 e^{- y}$ .

$e^{- y} \frac{d x}{d y} - x e^{- y} = 2 y e^{- y} + 3 e^{- y}$

Step 4

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

$\frac{d}{d y} [e^{- y} x] = 2 y e^{- y} + 3 e^{- y}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d y} [e^{- y} x] d y = \int 2 y e^{- y} + 3 e^{- y} d y$

Step 6

Integrate the left side.

$e^{- y} x = \int 2 y e^{- y} + 3 e^{- y} d y$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$e^{- y} x = \int 2 y e^{- y} d y + \int 3 e^{- y} d y$

Step 7.2

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

$e^{- y} x = 2 \int y e^{- y} d y + \int 3 e^{- y} d y$

Step 7.3

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

$e^{- y} x = 2 (y (- e^{- y}) - \int - e^{- y} d y) + \int 3 e^{- y} d y$

Step 7.4

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

$e^{- y} x = 2 (y (- e^{- y}) - - \int e^{- y} d y) + \int 3 e^{- y} d y$

Step 7.5

Simplify.

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

$e^{- y} x = 2 (y (- e^{- y}) + 1 \int e^{- y} d y) + \int 3 e^{- y} d y$

Step 7.5.2

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

$e^{- y} x = 2 (y (- e^{- y}) + \int e^{- y} d y) + \int 3 e^{- y} d y$

Step 7.6

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

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Step 7.6.1

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

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Step 7.6.1.1

Differentiate $- y$ .

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

Step 7.6.1.2

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

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

Step 7.6.1.3

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

$- 1 \cdot 1$

Step 7.6.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.6.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$e^{- y} x = 2 (y (- e^{- y}) + \int - e^{u_{1}} d u_{1}) + \int 3 e^{- y} d y$

Step 7.7

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

$e^{- y} x = 2 (y (- e^{- y}) - \int e^{u_{1}} d u_{1}) + \int 3 e^{- y} d y$

Step 7.8

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

$e^{- y} x = 2 (y (- e^{- y}) - (e^{u_{1}} + C)) + \int 3 e^{- y} d y$

Step 7.9

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$e^{- y} x = 2 (y (- e^{- y}) - (e^{u_{1}} + C)) + 3 \int e^{- y} d y$

Step 7.10

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

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Step 7.10.1

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

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Step 7.10.1.1

Differentiate $- y$ .

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

Step 7.10.1.2

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

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

Step 7.10.1.3

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

$- 1 \cdot 1$

Step 7.10.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.10.2

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

$e^{- y} x = 2 (y (- e^{- y}) - (e^{u_{1}} + C)) + 3 \int - e^{u_{2}} d u_{2}$

Step 7.11

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

$e^{- y} x = 2 (y (- e^{- y}) - (e^{u_{1}} + C)) + 3 (- \int e^{u_{2}} d u_{2})$

Step 7.12

Multiply $- 1$ by $3$ .

$e^{- y} x = 2 (y (- e^{- y}) - (e^{u_{1}} + C)) - 3 \int e^{u_{2}} d u_{2}$

Step 7.13

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

$e^{- y} x = 2 (y (- e^{- y}) - (e^{u_{1}} + C)) - 3 (e^{u_{2}} + C)$

Step 7.14

Simplify.

$e^{- y} x = 2 (- y e^{- y} - e^{u_{1}}) - 3 e^{u_{2}} + C$

Step 7.15

Substitute back in for each integration substitution variable.

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Step 7.15.1

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

$e^{- y} x = 2 (- y e^{- y} - e^{- y}) - 3 e^{u_{2}} + C$

Step 7.15.2

Replace all occurrences of $u_{2}$ with $- y$ .

$e^{- y} x = 2 (- y e^{- y} - e^{- y}) - 3 e^{- y} + C$

Step 8

Divide each term in $e^{- y} x = 2 (- y e^{- y} - e^{- y}) - 3 e^{- y} + C$ by $e^{- y}$ and simplify.

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Step 8.1

Divide each term in $e^{- y} x = 2 (- y e^{- y} - e^{- y}) - 3 e^{- y} + C$ by $e^{- y}$ .

$\frac{e^{- y} x}{e^{- y}} = \frac{2 (- y e^{- y} - e^{- y})}{e^{- y}} + \frac{- 3 e^{- y}}{e^{- y}} + \frac{C}{e^{- y}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{e^{- y} x}{e^{- y}} = \frac{2 (- y e^{- y} - e^{- y})}{e^{- y}} + \frac{- 3 e^{- y}}{e^{- y}} + \frac{C}{e^{- y}}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 (- y e^{- y} - e^{- y})}{e^{- y}} + \frac{- 3 e^{- y}}{e^{- y}} + \frac{C}{e^{- y}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

$x = \frac{2 (- y e^{- y} - e^{- y}) - 3 e^{- y} + C}{e^{- y}}$

Step 8.3.2

Simplify each term.

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Step 8.3.2.1

Apply the distributive property.

$x = \frac{2 (- y e^{- y}) + 2 (- e^{- y}) - 3 e^{- y} + C}{e^{- y}}$

Step 8.3.2.2

Multiply $- 1$ by $2$ .

$x = \frac{- 2 (y e^{- y}) + 2 (- e^{- y}) - 3 e^{- y} + C}{e^{- y}}$

Step 8.3.2.3

Multiply $- 1$ by $2$ .

$x = \frac{- 2 y e^{- y} - 2 e^{- y} - 3 e^{- y} + C}{e^{- y}}$

Step 8.3.3

Simplify terms.

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Step 8.3.3.1

Subtract $3 e^{- y}$ from $- 2 e^{- y}$ .

$x = \frac{- 2 y e^{- y} - 5 e^{- y} + C}{e^{- y}}$

Step 8.3.3.2

Factor $- 1$ out of $- 2 y e^{- y}$ .

$x = \frac{- (2 y e^{- y}) - 5 e^{- y} + C}{e^{- y}}$

Step 8.3.3.3

Factor $- 1$ out of $- 5 e^{- y}$ .

$x = \frac{- (2 y e^{- y}) - (5 e^{- y}) + C}{e^{- y}}$

Step 8.3.3.4

Factor $- 1$ out of $- (2 y e^{- y}) - (5 e^{- y})$ .

$x = \frac{- (2 y e^{- y} + 5 e^{- y}) + C}{e^{- y}}$

Step 8.3.3.5

Factor $- 1$ out of $C$ .

$x = \frac{- (2 y e^{- y} + 5 e^{- y}) - 1 (- C)}{e^{- y}}$

Step 8.3.3.6

Factor $- 1$ out of $- (2 y e^{- y} + 5 e^{- y}) - 1 (- C)$ .

$x = \frac{- (2 y e^{- y} + 5 e^{- y} - C)}{e^{- y}}$

Step 8.3.3.7

Simplify the expression.

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Step 8.3.3.7.1

Rewrite $- (2 y e^{- y} + 5 e^{- y} - C)$ as $- 1 (2 y e^{- y} + 5 e^{- y} - C)$ .

$x = \frac{- 1 (2 y e^{- y} + 5 e^{- y} - C)}{e^{- y}}$

Step 8.3.3.7.2

Move the negative in front of the fraction.

$x = - \frac{2 y e^{- y} + 5 e^{- y} - C}{e^{- y}}$