Calculus Examples

$(x - 4) \frac{d y}{d x} + y = 6 e^{x}$

Step 1

Check if the left side of the equation is the result of the derivative of the term $(x - 4) y$ .

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 4$ and $g (x) = y$ .

$(x - 4) \frac{d}{d x} [y] + y \frac{d}{d x} [x - 4]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$(x - 4) y' + y \frac{d}{d x} [x - 4]$

Step 1.3

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$(x - 4) y' + y (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 1.4

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

$(x - 4) y' + y (1 + \frac{d}{d x} [- 4])$

Step 1.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$(x - 4) y' + y (1 + 0)$

Step 1.6

Add $1$ and $0$ .

$(x - 4) y' + y \cdot 1$

Step 1.7

Substitute $\frac{d y}{d x}$ for $y'$ .

$(x - 4) \frac{d y}{d x} + y \cdot 1$

Step 1.8

Reorder $y$ and $1$ .

$(x - 4) \frac{d y}{d x} + 1 \cdot y$

Step 1.9

Multiply $y$ by $1$ .

$(x - 4) \frac{d y}{d x} + y$

Step 2

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

$\frac{d}{d x} [(x - 4) y] = 6 e^{x}$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [(x - 4) y] d x = \int 6 e^{x} d x$

Step 4

Integrate the left side.

$(x - 4) y = \int 6 e^{x} d x$

Step 5

Integrate the right side.

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Step 5.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$(x - 4) y = 6 \int e^{x} d x$

Step 5.2

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

$(x - 4) y = 6 (e^{x} + C)$

Step 5.3

Simplify.

$(x - 4) y = 6 e^{x} + C$

Step 6

Divide each term in $(x - 4) y = 6 e^{x} + C$ by $x - 4$ and simplify.

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Step 6.1

Divide each term in $(x - 4) y = 6 e^{x} + C$ by $x - 4$ .

$\frac{(x - 4) y}{x - 4} = \frac{6 e^{x}}{x - 4} + \frac{C}{x - 4}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $x - 4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{(x - 4) y}{x - 4} = \frac{6 e^{x}}{x - 4} + \frac{C}{x - 4}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 e^{x}}{x - 4} + \frac{C}{x - 4}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Combine the numerators over the common denominator.

$y = \frac{6 e^{x} + C}{x - 4}$