Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $y$ .

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.2.2.1

Factor $y$ out of $- y$ .

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

Step 1.2.2.2

Factor $y$ out of $2 y$ .

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

Step 1.2.2.3

Cancel the common factor.

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

Step 1.2.2.4

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

Simplify the answer.

Tap for more steps...

Step 2.3.4.1

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

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

Step 2.3.4.2

Simplify.

Tap for more steps...

Step 2.3.4.2.1

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

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

Step 2.3.4.2.2

Multiply $2$ by $2$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

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

Tap for more steps...

Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (- \frac{1}{4} x^{2} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Combine $x^{2}$ and $\frac{1}{4}$ .

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

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

Step 3.2.2.1.3.2

Factor $2$ out of $4$ .

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

Step 3.2.2.1.3.3

Cancel the common factor.

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

Step 3.2.2.1.3.4

Rewrite the expression.

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

Step 3.2.2.1.4

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 3.4

Simplify $\pm \sqrt{- \frac{x^{2}}{2} + 2 K}$ .

Tap for more steps...

Step 3.4.1

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Combine the numerators over the common denominator.

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

Step 3.4.4

Multiply $2$ by $2$ .

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

Step 3.4.5

Rewrite $\sqrt{\frac{- x^{2} + 4 K}{2}}$ as $\frac{\sqrt{- x^{2} + 4 K}}{\sqrt{2}}$ .

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

Step 3.4.6

Multiply $\frac{\sqrt{- x^{2} + 4 K}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{- x^{2} + 4 K}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.4.7

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.7.1

Multiply $\frac{\sqrt{- x^{2} + 4 K}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- x^{2} + 4 K} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- x^{2} + 4 K} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{- x^{2} + 4 K} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.4.7.5

Add $1$ and $1$ .

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

Step 3.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 3.4.7.6.1

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

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

Step 3.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{- x^{2} + 4 K} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.4.7.6.3

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

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

Step 3.4.7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.7.6.4.1

Cancel the common factor.

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

Step 3.4.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{- x^{2} + 4 K} \sqrt{2}}{2^{1}}$

Step 3.4.7.6.5

Evaluate the exponent.

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

Step 3.4.8

Combine using the product rule for radicals.

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

Step 3.4.9

Reorder factors in $\pm \frac{\sqrt{(- x^{2} + 4 K) \cdot 2}}{2}$ .

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

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$y = \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

Step 3.5.2

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

$y = - \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

Step 3.5.3

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

$y = \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

$y = - \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

$y = \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

$y = - \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

$y = \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

$y = - \frac{\sqrt{2 (- x^{2} + 4 K)}}{2}$

Step 4

Simplify the constant of integration.

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

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