Calculus Examples

$\sin (x) \frac{d y}{d x} + y \cos (x) = 1$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Factor $y$ out of $y \cos (x)$ .

$\sin (x) \frac{d y}{d x} + y (\cos (x)) = 1$

Step 1.2

Reorder $y$ and $\cos (x)$ .

$\sin (x) \frac{d y}{d x} + \cos (x) y = 1$

Step 2

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

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Step 2.1

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 $f (x) = \sin (x)$ and $g (x) = y$ .

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

Step 2.2

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

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

Step 2.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\sin (x) y' + y \cos (x)$

Step 2.4

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

$\sin (x) \frac{d y}{d x} + y \cos (x)$

Step 3

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

$\frac{d}{d x} [\sin (x) y] = 1$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [\sin (x) y] d x = \int d x$

Step 5

Integrate the left side.

$\sin (x) y = \int d x$

Step 6

Apply the constant rule.

$\sin (x) y = x + C$

Step 7

Divide each term in $\sin (x) y = x + C$ by $\sin (x)$ and simplify.

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Step 7.1

Divide each term in $\sin (x) y = x + C$ by $\sin (x)$ .

$\frac{\sin (x) y}{\sin (x)} = \frac{x}{\sin (x)} + \frac{C}{\sin (x)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\sin (x)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{\sin (x) y}{\sin (x)} = \frac{x}{\sin (x)} + \frac{C}{\sin (x)}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{\sin (x)} + \frac{C}{\sin (x)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Separate fractions.

$y = \frac{x}{1} \cdot \frac{1}{\sin (x)} + \frac{C}{\sin (x)}$

Step 7.3.1.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$y = \frac{x}{1} \csc (x) + \frac{C}{\sin (x)}$

Step 7.3.1.3

Divide $x$ by $1$ .

$y = x \csc (x) + \frac{C}{\sin (x)}$

Step 7.3.1.4

Separate fractions.

$y = x \csc (x) + \frac{C}{1} \cdot \frac{1}{\sin (x)}$

Step 7.3.1.5

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$y = x \csc (x) + \frac{C}{1} \csc (x)$

Step 7.3.1.6

Divide $C$ by $1$ .

$y = x \csc (x) + C \csc (x)$