Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Simplify each term.

Tap for more steps...

Step 1.1.2.1

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

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Move the negative in front of the fraction.

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

Step 1.2

Rewrite $\frac{y^{2}}{x^{2}}$ as ${(\frac{y}{x})}^{2}$ .

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

Step 2

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

$\frac{d y}{d x} = \sqrt{1 - V^{2}} + V$

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 $\frac{d y}{d x}$ .

$x \frac{d V}{d x} + V = \sqrt{1 - V^{2}} + V$

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

Simplify each term.

Tap for more steps...

Step 6.1.1.1.1

Rewrite $1$ as $1^{2}$ .

$x \frac{d V}{d x} + V = \sqrt{1^{2} - V^{2}} + V$

Step 6.1.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = V$ .

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

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.

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

Step 6.1.1.2.2

Combine the opposite terms in $\sqrt{(1 + V) (1 - V)} + V - V$ .

Tap for more steps...

Step 6.1.1.2.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = \sqrt{(1 + V) (1 - V)} + 0$

Step 6.1.1.2.2.2

Add $\sqrt{(1 + V) (1 - V)}$ and $0$ .

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

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = \sqrt{(1 + V) (1 - V)}$ by $x$ and simplify.

Tap for more steps...

Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \sqrt{(1 + V) (1 - V)}$ by $x$ .

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

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.

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

Step 6.1.1.3.2.1.2

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

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

Step 6.1.2

Multiply both sides by $\frac{1}{\sqrt{(1 + V) (1 - V)}}$ .

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

Step 6.1.3

Cancel the common factor of $\sqrt{(1 + V) (1 - V)}$ .

Tap for more steps...

Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Rewrite the expression.

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

Step 6.1.4

Rewrite the equation.

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

Step 6.2

Integrate both sides.

Tap for more steps...

Step 6.2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{(1 + V) (1 - V)}} d V = \int \frac{1}{x} d x$

Step 6.2.2

Integrate the left side.

Tap for more steps...

Step 6.2.2.1

Complete the square.

Tap for more steps...

Step 6.2.2.1.1

Simplify the expression.

Tap for more steps...

Step 6.2.2.1.1.1

Expand $(1 + V) (1 - V)$ using the FOIL Method.

Tap for more steps...

Step 6.2.2.1.1.1.1

Apply the distributive property.

$1 (1 - V) + V (1 - V)$

Step 6.2.2.1.1.1.2

Apply the distributive property.

$1 \cdot 1 + 1 (- V) + V (1 - V)$

Step 6.2.2.1.1.1.3

Apply the distributive property.

$1 \cdot 1 + 1 (- V) + V \cdot 1 + V (- V)$

Step 6.2.2.1.1.2

Simplify and combine like terms.

Tap for more steps...

Step 6.2.2.1.1.2.1

Simplify each term.

Tap for more steps...

Step 6.2.2.1.1.2.1.1

Multiply $1$ by $1$ .

$1 + 1 (- V) + V \cdot 1 + V (- V)$

Step 6.2.2.1.1.2.1.2

Multiply $- V$ by $1$ .

$1 - V + V \cdot 1 + V (- V)$

Step 6.2.2.1.1.2.1.3

Multiply $V$ by $1$ .

$1 - V + V + V (- V)$

Step 6.2.2.1.1.2.1.4

Rewrite using the commutative property of multiplication.

$1 - V + V - V \cdot V$

Step 6.2.2.1.1.2.1.5

Multiply $V$ by $V$ by adding the exponents.

Tap for more steps...

Step 6.2.2.1.1.2.1.5.1

Move $V$ .

$1 - V + V - (V \cdot V)$

Step 6.2.2.1.1.2.1.5.2

Multiply $V$ by $V$ .

$1 - V + V - V^{2}$

Step 6.2.2.1.1.2.2

Add $- V$ and $V$ .

$1 + 0 - V^{2}$

Step 6.2.2.1.1.2.3

Add $1$ and $0$ .

$1 - V^{2}$

Step 6.2.2.1.1.3

Reorder $1$ and $- V^{2}$ .

$- V^{2} + 1$

Step 6.2.2.1.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 0$

$c = 1$

Step 6.2.2.1.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 6.2.2.1.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 6.2.2.1.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 \cdot - 1}$

Step 6.2.2.1.4.2

Simplify the right side.

Tap for more steps...

Step 6.2.2.1.4.2.1

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 6.2.2.1.4.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 1}$

Step 6.2.2.1.4.2.1.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 6.2.2.1.4.2.2

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 6.2.2.1.4.2.3

Multiply $- 1$ by $0$ .

$d = 0$

Step 6.2.2.1.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 6.2.2.1.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 1 - \frac{0^{2}}{4 \cdot - 1}$

Step 6.2.2.1.5.2

Simplify the right side.

Tap for more steps...

Step 6.2.2.1.5.2.1

Simplify each term.

Tap for more steps...

Step 6.2.2.1.5.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = 1 - \frac{0}{4 \cdot - 1}$

Step 6.2.2.1.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 1 - \frac{0}{- 4}$

Step 6.2.2.1.5.2.1.3

Divide $0$ by $- 4$ .

$e = 1 - 0$

Step 6.2.2.1.5.2.1.4

Multiply $- 1$ by $0$ .

$e = 1 + 0$

Step 6.2.2.1.5.2.2

Add $1$ and $0$ .

$e = 1$

Step 6.2.2.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(V + 0)}^{2} + 1$ .

$\int \frac{1}{\sqrt{- {(V + 0)}^{2} + 1}} d V = \int \frac{1}{x} d x$

Step 6.2.2.2

Let $u = V + 0$ . Then $d u = d V$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 6.2.2.2.1

Let $u = V + 0$ . Find $\frac{d u}{d V}$ .

Tap for more steps...

Step 6.2.2.2.1.1

Differentiate $V + 0$ .

$\frac{d}{d V} [V + 0]$

Step 6.2.2.2.1.2

By the Sum Rule, the derivative of $V + 0$ with respect to $V$ is $\frac{d}{d V} [V] + \frac{d}{d V} [0]$ .

$\frac{d}{d V} [V] + \frac{d}{d V} [0]$

Step 6.2.2.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d V} [V^{n}]$ is $n V^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d V} [0]$

Step 6.2.2.2.1.4

Since $0$ is constant with respect to $V$ , the derivative of $0$ with respect to $V$ is $0$ .

$1 + 0$

Step 6.2.2.2.1.5

Add $1$ and $0$ .

$1$

Step 6.2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{\sqrt{- u^{2} + 1}} d u = \int \frac{1}{x} d x$

Step 6.2.2.3

Simplify the expression.

Tap for more steps...

Step 6.2.2.3.1

Rewrite $1$ as $1^{2}$ .

$\int \frac{1}{\sqrt{- u^{2} + 1^{2}}} d u = \int \frac{1}{x} d x$

Step 6.2.2.3.2

Reorder $- u^{2}$ and $1^{2}$ .

$\int \frac{1}{\sqrt{1^{2} - u^{2}}} d u = \int \frac{1}{x} d x$

Step 6.2.2.4

The integral of $\frac{1}{\sqrt{1^{2} - u^{2}}}$ with respect to $u$ is $\arcsin (u)$

$\arcsin (u) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.2.5

Replace all occurrences of $u$ with $V + 0$ .

$\arcsin (V + 0) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.2.6

Add $V$ and $0$ .

$\arcsin (V) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.3

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

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

Step 6.2.4

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

$\arcsin (V) = \ln (| x |) + C$

Step 6.3

Take the inverse arcsine of both sides of the equation to extract $V$ from inside the arcsine.

$V = \sin (\ln (| x |) + C)$

Step 7

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

$\frac{y}{x} = \sin (\ln (| x |) + C)$

Step 8

Solve $\frac{y}{x} = \sin (\ln (| x |) + C)$ for $y$ .

Tap for more steps...

Step 8.1

Multiply both sides by $x$ .

$\frac{y}{x} x = \sin (\ln (| x |) + C) x$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Simplify the left side.

Tap for more steps...

Step 8.2.1.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 8.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = \sin (\ln (| x |) + C) x$

Step 8.2.1.1.2

Rewrite the expression.

$y = \sin (\ln (| x |) + C) x$

Step 8.2.2

Simplify the right side.

Tap for more steps...

Step 8.2.2.1

Reorder factors in $\sin (\ln (| x |) + C) x$ .

$y = x \sin (\ln (| x |) + C)$