Calculus Examples

$\sin (x) \frac{d y}{d x} + (\cos (x)) y = 5 \sin (x^{2})$

Step 1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

$\frac{d}{d x} [\sin (x) y] = 5 \sin (x^{2})$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [\sin (x) y] d x = \int 5 \sin (x^{2}) d x$

Step 4

Integrate the left side.

$\sin (x) y = \int 5 \sin (x^{2}) d x$

Step 5

Integrate the right side.

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Step 5.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\sin (x) y = 5 \int \sin (x^{2}) d x$

Step 5.2

$\int \sin (x^{2}) d x$ is a special integral. The integral is the Fresnel integral function.

$\sin (x) y = 5 (S (x) + C)$

Step 5.3

Simplify.

$\sin (x) y = 5 S (x) + C$

Step 6

Divide each term in $\sin (x) y = 5 S x + C$ by $\sin (x)$ and simplify.

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Step 6.1

Divide each term in $\sin (x) y = 5 S x + C$ by $\sin (x)$ .

$\frac{\sin (x) y}{\sin (x)} = \frac{5 S x}{\sin (x)} + \frac{C}{\sin (x)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

$\frac{\sin (x) y}{\sin (x)} = \frac{5 S x}{\sin (x)} + \frac{C}{\sin (x)}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 S x}{\sin (x)} + \frac{C}{\sin (x)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Separate fractions.

$y = \frac{5 S x}{\sin (x)} + \frac{C}{1} \cdot \frac{1}{\sin (x)}$

Step 6.3.1.2

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

$y = \frac{5 S x}{\sin (x)} + \frac{C}{1} \csc (x)$

Step 6.3.1.3

Divide $C$ by $1$ .

$y = \frac{5 S x}{\sin (x)} + C \csc (x)$