Calculus Examples

$d x - x^{2} d y = 0$

Step 1

Subtract $d x$ from both sides of the equation.

$- x^{2} d y = - d x$

Step 2

Multiply both sides by $\frac{1}{x^{2}}$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

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

Step 4.2

Apply the constant rule.

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

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

Apply basic rules of exponents.

Tap for more steps...

Step 4.3.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.3.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.3

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

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

Step 4.3.4

Simplify the answer.

Tap for more steps...

Step 4.3.4.1

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

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

Step 4.3.4.2

Simplify.

Tap for more steps...

Step 4.3.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.4.2.2

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

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

Step 4.4

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

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

Step 5

Divide each term in $- y = \frac{1}{x} + K$ by $- 1$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $- y = \frac{1}{x} + K$ by $- 1$ .

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

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $y$ by $1$ .

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

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

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

$y = - 1 \cdot \frac{1}{x} + \frac{K}{- 1}$

Step 5.3.1.2

Rewrite $- 1 \cdot \frac{1}{x}$ as $- \frac{1}{x}$ .

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

Step 5.3.1.3

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

$y = - \frac{1}{x} - 1 \cdot K$

Step 5.3.1.4

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

$y = - \frac{1}{x} - K$

Step 6

Simplify the constant of integration.

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