Calculus Examples

$\frac{d y}{d x} + 2 y = 50$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 2$ .

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Step 1.1

Set up the integration.

$e^{\int 2 d x}$

Step 1.2

Apply the constant rule.

$e^{2 x + C}$

Step 1.3

Remove the constant of integration.

$e^{2 x}$

Step 2

Multiply each term by the integrating factor $e^{2 x}$ .

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Step 2.1

Multiply each term by $e^{2 x}$ .

$e^{2 x} \frac{d y}{d x} + e^{2 x} (2 y) = e^{2 x} \cdot 50$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{2 x} \frac{d y}{d x} + 2 e^{2 x} y = e^{2 x} \cdot 50$

Step 2.3

Move $50$ to the left of $e^{2 x}$ .

$e^{2 x} \frac{d y}{d x} + 2 e^{2 x} y = 50 \cdot e^{2 x}$

Step 2.4

Reorder factors in $e^{2 x} \frac{d y}{d x} + 2 e^{2 x} y = 50 e^{2 x}$ .

$e^{2 x} \frac{d y}{d x} + 2 y e^{2 x} = 50 e^{2 x}$

Step 3

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

$\frac{d}{d x} [e^{2 x} y] = 50 e^{2 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{2 x} y] d x = \int 50 e^{2 x} d x$

Step 5

Integrate the left side.

$e^{2 x} y = \int 50 e^{2 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $50$ is constant with respect to $x$ , move $50$ out of the integral.

$e^{2 x} y = 50 \int e^{2 x} d x$

Step 6.2

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 6.2.1

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

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Step 6.2.1.1

Differentiate $2 x$ .

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

Step 6.2.1.2

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 6.2.1.3

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 6.2.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2.2

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

$e^{2 x} y = 50 \int e^{u} \frac{1}{2} d u$

Step 6.3

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

$e^{2 x} y = 50 \int \frac{e^{u}}{2} d u$

Step 6.4

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

$e^{2 x} y = 50 (\frac{1}{2} \int e^{u} d u)$

Step 6.5

Simplify.

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Step 6.5.1

Combine $\frac{1}{2}$ and $50$ .

$e^{2 x} y = \frac{50}{2} \int e^{u} d u$

Step 6.5.2

Cancel the common factor of $50$ and $2$ .

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Step 6.5.2.1

Factor $2$ out of $50$ .

$e^{2 x} y = \frac{2 \cdot 25}{2} \int e^{u} d u$

Step 6.5.2.2

Cancel the common factors.

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Step 6.5.2.2.1

Factor $2$ out of $2$ .

$e^{2 x} y = \frac{2 \cdot 25}{2 (1)} \int e^{u} d u$

Step 6.5.2.2.2

Cancel the common factor.

$e^{2 x} y = \frac{2 \cdot 25}{2 \cdot 1} \int e^{u} d u$

Step 6.5.2.2.3

Rewrite the expression.

$e^{2 x} y = \frac{25}{1} \int e^{u} d u$

Step 6.5.2.2.4

Divide $25$ by $1$ .

$e^{2 x} y = 25 \int e^{u} d u$

Step 6.6

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

$e^{2 x} y = 25 (e^{u} + C)$

Step 6.7

Simplify.

$e^{2 x} y = 25 e^{u} + C$

Step 6.8

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

$e^{2 x} y = 25 e^{2 x} + C$

Step 7

Divide each term in $e^{2 x} y = 25 e^{2 x} + C$ by $e^{2 x}$ and simplify.

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Step 7.1

Divide each term in $e^{2 x} y = 25 e^{2 x} + C$ by $e^{2 x}$ .

$\frac{e^{2 x} y}{e^{2 x}} = \frac{25 e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{2 x} y}{e^{2 x}} = \frac{25 e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{25 e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

$y = \frac{25 e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.2

Divide $25$ by $1$ .

$y = 25 + \frac{C}{e^{2 x}}$