Calculus Examples

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

Step 1

Rewrite the differential equation as a function of $\frac{y}{x}$ .

Tap for more steps...

Step 1.1

Split $\frac{x^{2} + y^{2}}{x \cdot y}$ and simplify.

Tap for more steps...

Step 1.1.1

Split the fraction $\frac{x^{2} + y^{2}}{x \cdot y}$ into two fractions.

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

Step 1.1.2

Simplify each term.

Tap for more steps...

Step 1.1.2.1

Cancel the common factor of $x^{2}$ and $x$ .

Tap for more steps...

Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.1.2.1

Factor $x$ out of $x \cdot y$ .

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

Step 1.1.2.1.2.2

Cancel the common factor.

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

Step 1.1.2.1.2.3

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $y^{2}$ and $y$ .

Tap for more steps...

Step 1.1.2.2.1

Factor $y$ out of $y^{2}$ .

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

Step 1.1.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.2.2.1

Factor $y$ out of $x \cdot y$ .

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

Step 1.1.2.2.2.2

Cancel the common factor.

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

Step 1.1.2.2.2.3

Rewrite the expression.

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

Step 1.2

Rewrite $\frac{x}{y}$ as ${(\frac{y}{x})}^{- 1}$ .

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

Step 2

Let $V = \frac{y}{x}$ . Substitute $V$ for $\frac{y}{x}$ .

$\frac{d y}{d x} = V^{- 1} + V$

Step 3

Solve $V = \frac{y}{x}$ for $y$ .

$y = V x$

Step 4

Use the product rule to find the derivative of $y = V x$ with respect to $x$ .

$\frac{d y}{d x} = x \frac{d V}{d x} + V$

Step 5

Substitute $x \frac{d V}{d x} + V$ for $\frac{d y}{d x}$ .

$x \frac{d V}{d x} + V = V^{- 1} + V$

Step 6

Solve the substituted differential equation.

Tap for more steps...

Step 6.1

Separate the variables.

Tap for more steps...

Step 6.1.1

Solve for $\frac{d V}{d x}$ .

Tap for more steps...

Step 6.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x \frac{d V}{d x} + V = \frac{1}{V} + V$

Step 6.1.1.2

Move all terms not containing $\frac{d V}{d x}$ to the right side of the equation.

Tap for more steps...

Step 6.1.1.2.1

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = \frac{1}{V} + V - V$

Step 6.1.1.2.2

Combine the opposite terms in $\frac{1}{V} + V - V$ .

Tap for more steps...

Step 6.1.1.2.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = \frac{1}{V} + 0$

Step 6.1.1.2.2.2

Add $\frac{1}{V}$ and $0$ .

$x \frac{d V}{d x} = \frac{1}{V}$

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = \frac{1}{V}$ by $x$ and simplify.

Tap for more steps...

Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \frac{1}{V}$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{1}{V}}{x}$

Step 6.1.1.3.2

Simplify the left side.

Tap for more steps...

Step 6.1.1.3.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.1.1.3.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{1}{V}}{x}$

Step 6.1.1.3.2.1.2

Divide $\frac{d V}{d x}$ by $1$ .

$\frac{d V}{d x} = \frac{\frac{1}{V}}{x}$

Step 6.1.1.3.3

Simplify the right side.

Tap for more steps...

Step 6.1.1.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d V}{d x} = \frac{1}{V} \cdot \frac{1}{x}$

Step 6.1.1.3.3.2

Multiply $\frac{1}{V}$ by $\frac{1}{x}$ .

$\frac{d V}{d x} = \frac{1}{V x}$

Step 6.1.2

Regroup factors.

$\frac{d V}{d x} = \frac{1}{V} \cdot \frac{1}{x}$

Step 6.1.3

Multiply both sides by $V$ .

$V \frac{d V}{d x} = V (\frac{1}{V} \cdot \frac{1}{x})$

Step 6.1.4

Simplify.

Tap for more steps...

Step 6.1.4.1

Multiply $\frac{1}{V}$ by $\frac{1}{x}$ .

$V \frac{d V}{d x} = V \frac{1}{V x}$

Step 6.1.4.2

Cancel the common factor of $V$ .

Tap for more steps...

Step 6.1.4.2.1

Factor $V$ out of $V x$ .

$V \frac{d V}{d x} = V \frac{1}{V (x)}$

Step 6.1.4.2.2

Cancel the common factor.

$V \frac{d V}{d x} = V \frac{1}{V x}$

Step 6.1.4.2.3

Rewrite the expression.

$V \frac{d V}{d x} = \frac{1}{x}$

Step 6.1.5

Rewrite the equation.

$V d V = \frac{1}{x} d x$

Step 6.2

Integrate both sides.

Tap for more steps...

Step 6.2.1

Set up an integral on each side.

$\int V d V = \int \frac{1}{x} d x$

Step 6.2.2

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

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

Step 6.2.3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{2} V^{2} + C_{1} = \ln (| x |) + C_{2}$

Step 6.2.4

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

$\frac{1}{2} V^{2} = \ln (| x |) + C$

Step 6.3

Solve for $V$ .

Tap for more steps...

Step 6.3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} V^{2}) = 2 (\ln (| x |) + C)$

Step 6.3.2

Simplify both sides of the equation.

Tap for more steps...

Step 6.3.2.1

Simplify the left side.

Tap for more steps...

Step 6.3.2.1.1

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

Tap for more steps...

Step 6.3.2.1.1.1

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

$2 \frac{V^{2}}{2} = 2 (\ln (| x |) + C)$

Step 6.3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.1.1.2.1

Cancel the common factor.

$2 \frac{V^{2}}{2} = 2 (\ln (| x |) + C)$

Step 6.3.2.1.1.2.2

Rewrite the expression.

$V^{2} = 2 (\ln (| x |) + C)$

Step 6.3.2.2

Simplify the right side.

Tap for more steps...

Step 6.3.2.2.1

Apply the distributive property.

$V^{2} = 2 \ln (| x |) + 2 C$

Step 6.3.3

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

$V^{2} = \ln ({| x |}^{2}) + 2 C$

Step 6.3.4

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

$V = \pm \sqrt{\ln ({| x |}^{2}) + 2 C}$

Step 6.3.5

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$V = \pm \sqrt{\ln (x^{2}) + 2 C}$

Step 6.3.6

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

Tap for more steps...

Step 6.3.6.1

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

$V = \sqrt{\ln (x^{2}) + 2 C}$

Step 6.3.6.2

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

$V = - \sqrt{\ln (x^{2}) + 2 C}$

Step 6.3.6.3

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

$V = \sqrt{\ln (x^{2}) + 2 C}$

$V = - \sqrt{\ln (x^{2}) + 2 C}$

$V = \sqrt{\ln (x^{2}) + 2 C}$

$V = - \sqrt{\ln (x^{2}) + 2 C}$

$V = \sqrt{\ln (x^{2}) + 2 C}$

$V = - \sqrt{\ln (x^{2}) + 2 C}$

Step 6.4

Simplify the constant of integration.

$V = \sqrt{\ln (x^{2}) + C}$

$V = - \sqrt{\ln (x^{2}) + C}$

$V = \sqrt{\ln (x^{2}) + C}$

$V = - \sqrt{\ln (x^{2}) + C}$

Step 7

Substitute $\frac{y}{x}$ for $V$ .

$\frac{y}{x} = \sqrt{\ln (x^{2}) + C}, - \sqrt{\ln (x^{2}) + C}$

Step 8

Solve $\frac{y}{x} = \sqrt{\ln (x^{2}) + C}$ for $y$ .

Tap for more steps...

Step 8.1

Rewrite.

$\frac{y}{x} = \sqrt{\ln (x^{2}) + C}$

Step 8.2

Multiply both sides by $x$ .

$\frac{y}{x} x = \sqrt{\ln (x^{2}) + C} x$

Step 8.3

Simplify the left side.

Tap for more steps...

Step 8.3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 8.3.1.1

Cancel the common factor.

$\frac{y}{x} x = \sqrt{\ln (x^{2}) + C} x$

Step 8.3.1.2

Rewrite the expression.

$y = \sqrt{\ln (x^{2}) + C} x$

Step 9

Solve $\frac{y}{x} = - \sqrt{\ln (x^{2}) + C}$ for $y$ .

Tap for more steps...

Step 9.1

Rewrite.

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

Step 9.2

Multiply both sides by $x$ .

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

Step 9.3

Simplify the left side.

Tap for more steps...

Step 9.3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 9.3.1.1

Cancel the common factor.

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

Step 9.3.1.2

Rewrite the expression.

$y = - \sqrt{\ln (x^{2}) + C} x$

Step 10

List the solutions.

$y = \sqrt{\ln (x^{2}) + C} x, y = - \sqrt{\ln (x^{2}) + C} x$