Calculus Examples

$\frac{d y}{d x} = \frac{\sqrt{x}}{3 y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{\sqrt{x}}{3 y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $3 y$ .

$y \frac{d y}{d x} = y \frac{\sqrt{x}}{y \cdot 3}$

Step 1.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{\sqrt{x}}{y \cdot 3}$

Step 1.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{\sqrt{x}}{3}$

Step 1.3

Rewrite the equation.

$y d y = \frac{\sqrt{x}}{3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{\sqrt{x}}{3} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{\sqrt{x}}{3} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} \int \sqrt{x} d x$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} \int x^{\frac{1}{2}} d x$

Step 2.3.3

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} (\frac{2}{3} x^{\frac{3}{2}} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $\frac{1}{3} (\frac{2}{3} x^{\frac{3}{2}} + C_{2})$ as $\frac{1}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $\frac{1}{3}$ by $\frac{2}{3}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{2}{3 \cdot 3} x^{\frac{3}{2}} + C_{2}$

Step 2.3.4.2.2

Multiply $3$ by $3$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{2}{9} x^{\frac{3}{2}} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{2}{9} x^{\frac{3}{2}} + K$