Calculus Examples

$\frac{d y}{d x} = - \frac{4 y^{2}}{x^{3}}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{1}{y^{2}} (- \frac{4 y^{2}}{x^{3}})$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{2}} \frac{d y}{d x} = - \frac{1}{y^{2}} \cdot \frac{4 y^{2}}{x^{3}}$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

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

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{- 1}{y^{2}} \cdot \frac{4 y^{2}}{x^{3}}$

Step 1.2.2.2

Factor $y^{2}$ out of $4 y^{2}$ .

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{- 1}{y^{2}} \cdot \frac{y^{2} \cdot 4}{x^{3}}$

Step 1.2.2.3

Cancel the common factor.

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{- 1}{y^{2}} \cdot \frac{y^{2} \cdot 4}{x^{3}}$

Step 1.2.2.4

Rewrite the expression.

$\frac{1}{y^{2}} \frac{d y}{d x} = - \frac{4}{x^{3}}$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{2}} d y = - \frac{4}{x^{3}} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{2}} d y = \int - \frac{4}{x^{3}} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1.1

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

$\int {(y^{2})}^{- 1} d y = \int - \frac{4}{x^{3}} d x$

Step 2.2.1.2

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

Tap for more steps...

Step 2.2.1.2.1

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

$\int y^{2 \cdot - 1} d y = \int - \frac{4}{x^{3}} d x$

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

$\int y^{- 2} d y = \int - \frac{4}{x^{3}} d x$

Step 2.2.2

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

$- y^{- 1} + C_{1} = \int - \frac{4}{x^{3}} d x$

Step 2.2.3

Rewrite $- y^{- 1} + C_{1}$ as $- \frac{1}{y} + C_{1}$ .

$- \frac{1}{y} + C_{1} = \int - \frac{4}{x^{3}} d x$

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.

$- \frac{1}{y} + C_{1} = - \int \frac{4}{x^{3}} d x$

Step 2.3.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$- \frac{1}{y} + C_{1} = - (4 \int \frac{1}{x^{3}} d x)$

Step 2.3.3

Simplify the expression.

Tap for more steps...

Step 2.3.3.1

Multiply $4$ by $- 1$ .

$- \frac{1}{y} + C_{1} = - 4 \int \frac{1}{x^{3}} d x$

Step 2.3.3.2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{y} + C_{1} = - 4 \int {(x^{3})}^{- 1} d x$

Step 2.3.3.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

Tap for more steps...

Step 2.3.3.3.1

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

$- \frac{1}{y} + C_{1} = - 4 \int x^{3 \cdot - 1} d x$

Step 2.3.3.3.2

Multiply $3$ by $- 1$ .

$- \frac{1}{y} + C_{1} = - 4 \int x^{- 3} d x$

Step 2.3.4

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

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

Step 2.3.5

Simplify the answer.

Tap for more steps...

Step 2.3.5.1

Simplify.

Tap for more steps...

Step 2.3.5.1.1

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

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

Step 2.3.5.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.5.2

Simplify.

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

Step 2.3.5.3

Simplify.

Tap for more steps...

Step 2.3.5.3.1

Multiply $- 1$ by $- 4$ .

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

Step 2.3.5.3.2

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

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

Step 2.3.5.3.3

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.3.5.3.3.1

Factor $2$ out of $4$ .

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

Step 2.3.5.3.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.5.3.3.2.1

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

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

Step 2.3.5.3.3.2.2

Cancel the common factor.

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

Step 2.3.5.3.3.2.3

Rewrite the expression.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, x^{2}, 1$

Step 3.1.2

Since $y, x^{2}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $y^{1}, x^{2}$ .

Step 3.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.1.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 3.1.6

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 3.1.7

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 3.1.8

The LCM of $y^{1}, x^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y \cdot x \cdot x$

Step 3.1.9

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.1.9.1

Move $x$ .

$y \cdot (x \cdot x)$

Step 3.1.9.2

Multiply $x$ by $x$ .

$y \cdot x^{2}$

$y x^{2}$

Step 3.2

Multiply each term in $- \frac{1}{y} = \frac{2}{x^{2}} + K$ by $y x^{2}$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $- \frac{1}{y} = \frac{2}{x^{2}} + K$ by $y x^{2}$ .

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Step 3.2.2.1.2

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

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

Step 3.2.2.1.3

Cancel the common factor.

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

Step 3.2.2.1.4

Rewrite the expression.

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

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

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

Tap for more steps...

Step 3.2.3.1.1

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

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

Step 3.2.3.1.2

Cancel the common factor.

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

Step 3.2.3.1.3

Rewrite the expression.

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

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

Step 3.3

Solve the equation.

Tap for more steps...

Step 3.3.1

Rewrite the equation as $2 y + K y x^{2} = - x^{2}$ .

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

Step 3.3.2

Factor $y$ out of $2 y + K y x^{2}$ .

Tap for more steps...

Step 3.3.2.1

Factor $y$ out of $2 y$ .

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Factor $y$ out of $y \cdot 2 + y (K x^{2})$ .

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

Step 3.3.3

Divide each term in $y (2 + K x^{2}) = - x^{2}$ by $2 + K x^{2}$ and simplify.

Tap for more steps...

Step 3.3.3.1

Divide each term in $y (2 + K x^{2}) = - x^{2}$ by $2 + K x^{2}$ .

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

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

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

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.3.1

Move the negative in front of the fraction.

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