Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

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

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$y d y = (3 x^{2} + 2) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 2.3.5.2

Simplify.

$\frac{1}{2} y^{2} + C_{1} = x^{3} + 2 x + C_{4}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (x^{3} + 2 x + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (x^{3} + 2 x + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{2} = 2 x^{3} + 2 (2 x) + 2 K$

Step 3.2.2.1.2

Multiply $2$ by $2$ .

$y^{2} = 2 x^{3} + 4 x + 2 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{2 x^{3} + 4 x + 2 K}$

Step 3.4

Factor $2$ out of $2 x^{3} + 4 x + 2 K$ .

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Step 3.4.1

Factor $2$ out of $2 x^{3}$ .

$y = \pm \sqrt{2 (x^{3}) + 4 x + 2 K}$

Step 3.4.2

Factor $2$ out of $4 x$ .

$y = \pm \sqrt{2 (x^{3}) + 2 (2 x) + 2 K}$

Step 3.4.3

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (x^{3}) + 2 (2 x) + 2 K}$

Step 3.4.4

Factor $2$ out of $2 (x^{3}) + 2 (2 x)$ .

$y = \pm \sqrt{2 (x^{3} + 2 x) + 2 K}$

Step 3.4.5

Factor $2$ out of $2 (x^{3} + 2 x) + 2 K$ .

$y = \pm \sqrt{2 (x^{3} + 2 x + K)}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (x^{3} + 2 x + K)}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (x^{3} + 2 x + K)}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (x^{3} + 2 x + K)}$

$y = - \sqrt{2 (x^{3} + 2 x + K)}$

$y = \sqrt{2 (x^{3} + 2 x + K)}$

$y = - \sqrt{2 (x^{3} + 2 x + K)}$

$y = \sqrt{2 (x^{3} + 2 x + K)}$

$y = - \sqrt{2 (x^{3} + 2 x + K)}$