Calculus Examples

$2 y \frac{d y}{d x} = - 4 x$

Step 1

Rewrite the equation.

$2 y d y = - 4 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 y d y = \int - 4 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$2 \int y d y = \int - 4 x d x$

Step 2.2.2

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

$2 (\frac{1}{2} y^{2} + C_{1}) = \int - 4 x d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $2 (\frac{1}{2} y^{2} + C_{1})$ as $2 (\frac{1}{2}) y^{2} + C_{1}$ .

$2 (\frac{1}{2}) y^{2} + C_{1} = \int - 4 x d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

$\frac{2}{2} y^{2} + C_{1} = \int - 4 x d x$

Step 2.2.3.2.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.2.1

Cancel the common factor.

$\frac{2}{2} y^{2} + C_{1} = \int - 4 x d x$

Step 2.2.3.2.2.2

Rewrite the expression.

$1 y^{2} + C_{1} = \int - 4 x d x$

Step 2.2.3.2.3

Multiply $y^{2}$ by $1$ .

$y^{2} + C_{1} = \int - 4 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$y^{2} + C_{1} = - 4 \int x d x$

Step 2.3.2

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

$y^{2} + C_{1} = - 4 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- 4 (\frac{1}{2} x^{2} + C_{2})$ as $- 4 (\frac{1}{2}) x^{2} + C_{2}$ .

$y^{2} + C_{1} = - 4 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $- 4$ and $\frac{1}{2}$ .

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

Step 2.3.3.2.2

Cancel the common factor of $- 4$ and $2$ .

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Step 2.3.3.2.2.1

Factor $2$ out of $- 4$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $- 2$ by $1$ .

$y^{2} + C_{1} = - 2 x^{2} + C_{2}$

Step 2.4

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

$y^{2} = - 2 x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

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

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

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

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

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