Calculus Examples

$\frac{d y}{d x} = \frac{x}{49 y}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{x}{49 y}$

Step 1.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.2.1

Factor $y$ out of $49 y$ .

$y \frac{d y}{d x} = y \frac{x}{y \cdot 49}$

Step 1.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x}{y \cdot 49}$

Step 1.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{x}{49}$

Step 1.3

Rewrite the equation.

$y d y = \frac{x}{49} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{x}{49} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x}{49} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $\frac{1}{49}$ is constant with respect to $x$ , move $\frac{1}{49}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{49} \int x d x$

Step 2.3.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{49} (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

Tap for more steps...

Step 2.3.3.1

Rewrite $\frac{1}{49} (\frac{1}{2} x^{2} + C_{2})$ as $\frac{1}{49} \cdot \frac{1}{2} x^{2} + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{49} \cdot \frac{1}{2} x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

Tap for more steps...

Step 2.3.3.2.1

Multiply $\frac{1}{49}$ by $\frac{1}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{49 \cdot 2} x^{2} + C_{2}$

Step 2.3.3.2.2

Multiply $49$ by $2$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{98} x^{2} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{1}{98} x^{2} + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{98} x^{2} + K)$

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 $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\frac{1}{98} x^{2} + K)$

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.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{98} x^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{98} x^{2} + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (\frac{1}{98} x^{2} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Combine $\frac{1}{98}$ and $x^{2}$ .

$y^{2} = 2 (\frac{x^{2}}{98} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{x^{2}}{98} + 2 K$

Step 3.2.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.3.1

Factor $2$ out of $98$ .

$y^{2} = 2 \frac{x^{2}}{2 (49)} + 2 K$

Step 3.2.2.1.3.2

Cancel the common factor.

$y^{2} = 2 \frac{x^{2}}{2 \cdot 49} + 2 K$

Step 3.2.2.1.3.3

Rewrite the expression.

$y^{2} = \frac{x^{2}}{49} + 2 K$

Step 3.3

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

$y = \pm \sqrt{\frac{x^{2}}{49} + 2 K}$

Step 3.4

Simplify $\pm \sqrt{\frac{x^{2}}{49} + 2 K}$ .

Tap for more steps...

Step 3.4.1

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$y = \pm \sqrt{\frac{x^{2}}{49} + 2 K \cdot \frac{49}{49}}$

Step 3.4.2

Combine $2 K$ and $\frac{49}{49}$ .

$y = \pm \sqrt{\frac{x^{2}}{49} + \frac{2 K \cdot 49}{49}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} + 2 K \cdot 49}{49}}$

Step 3.4.4

Multiply $49$ by $2$ .

$y = \pm \sqrt{\frac{x^{2} + 98 K}{49}}$

Step 3.4.5

Rewrite $\frac{x^{2} + 98 K}{49}$ as ${(\frac{1}{7})}^{2} (x^{2} + 98 K)$ .

Tap for more steps...

Step 3.4.5.1

Factor the perfect power $1^{2}$ out of $x^{2} + 98 K$ .

$y = \pm \sqrt{\frac{1^{2} (x^{2} + 98 K)}{49}}$

Step 3.4.5.2

Factor the perfect power $7^{2}$ out of $49$ .

$y = \pm \sqrt{\frac{1^{2} (x^{2} + 98 K)}{7^{2} \cdot 1}}$

Step 3.4.5.3

Rearrange the fraction $\frac{1^{2} (x^{2} + 98 K)}{7^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{7})}^{2} (x^{2} + 98 K)}$

Step 3.4.6

Pull terms out from under the radical.

$y = \pm \frac{1}{7} \sqrt{x^{2} + 98 K}$

Step 3.4.7

Combine $\frac{1}{7}$ and $\sqrt{x^{2} + 98 K}$ .

$y = \pm \frac{\sqrt{x^{2} + 98 K}}{7}$

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.

$y = \frac{\sqrt{x^{2} + 98 K}}{7}$

Step 3.5.2

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

$y = - \frac{\sqrt{x^{2} + 98 K}}{7}$

Step 3.5.3

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

$y = \frac{\sqrt{x^{2} + 98 K}}{7}$

$y = - \frac{\sqrt{x^{2} + 98 K}}{7}$

$y = \frac{\sqrt{x^{2} + 98 K}}{7}$

$y = - \frac{\sqrt{x^{2} + 98 K}}{7}$

$y = \frac{\sqrt{x^{2} + 98 K}}{7}$

$y = - \frac{\sqrt{x^{2} + 98 K}}{7}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{x^{2} + K}}{7}$

$y = - \frac{\sqrt{x^{2} + K}}{7}$