Calculus Examples

$\frac{d y}{d x} = x \sqrt{1 - y^{2}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sqrt{1 - y^{2}}}$ .

$\frac{1}{\sqrt{1 - y^{2}}} \frac{d y}{d x} = \frac{1}{\sqrt{1 - y^{2}}} (x \sqrt{1 - y^{2}})$

Step 1.2

Cancel the common factor of $\sqrt{1 - y^{2}}$ .

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Step 1.2.1

Factor $\sqrt{1 - y^{2}}$ out of $x \sqrt{1 - y^{2}}$ .

$\frac{1}{\sqrt{1 - y^{2}}} \frac{d y}{d x} = \frac{1}{\sqrt{1 - y^{2}}} (\sqrt{1 - y^{2}} x)$

Step 1.2.2

Cancel the common factor.

$\frac{1}{\sqrt{1 - y^{2}}} \frac{d y}{d x} = \frac{1}{\sqrt{1 - y^{2}}} (\sqrt{1 - y^{2}} x)$

Step 1.2.3

Rewrite the expression.

$\frac{1}{\sqrt{1 - y^{2}}} \frac{d y}{d x} = x$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{1 - y^{2}}} d y = x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{1 - y^{2}}} d y = \int x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Rewrite $1$ as $1^{2}$ .

$\int \frac{1}{\sqrt{1^{2} - y^{2}}} d y = \int x d x$

Step 2.2.2

The integral of $\frac{1}{\sqrt{1^{2} - y^{2}}}$ with respect to $y$ is $\arcsin (y)$

$\arcsin (y) + C_{1} = \int x d x$

Step 2.3

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\arcsin (y) + C_{1} = \frac{1}{2} x^{2} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\arcsin (y) = \frac{1}{2} x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Take the inverse arcsine of both sides of the equation to extract $y$ from inside the arcsine.

$y = \sin (\frac{1}{2} x^{2} + K)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y = \sin (\frac{x^{2}}{2} + K)$