Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

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

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

Step 1.2.3

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

Tap for more steps...

Step 1.2.3.1

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

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

Step 1.2.3.2

Cancel the common factor.

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

Step 1.2.3.3

Rewrite the expression.

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

Step 1.3

Remove unnecessary parentheses.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

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

Rewrite $\frac{1}{y^{2}}$ as ${(y^{2})}^{- 1}$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

Tap for more steps...

Step 2.2.3.1

Multiply $y^{- 2}$ by $y^{2}$ by adding the exponents.

Tap for more steps...

Step 2.2.3.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.3.1.2

Add $- 2$ and $2$ .

$\int y^{0} + y^{- 2} \cdot - 1 d y = \int 4 x d x$

Step 2.2.3.2

Simplify $y^{0}$ .

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

Step 2.2.3.3

Move $- 1$ to the left of $y^{- 2}$ .

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

Step 2.2.3.4

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

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

Step 2.2.4

Split the single integral into multiple integrals.

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

Step 2.2.5

Apply the constant rule.

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

Simplify.

Tap for more steps...

Step 2.2.8.1

Simplify.

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

Step 2.2.8.2

Simplify.

Tap for more steps...

Step 2.2.8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.8.2.2

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

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

$y + \frac{1}{y} + C_{3} = 4 (\frac{1}{2} x^{2} + C_{4})$

Step 2.3.3

Simplify the answer.

Tap for more steps...

Step 2.3.3.1

Rewrite $4 (\frac{1}{2} x^{2} + C_{4})$ as $4 (\frac{1}{2}) x^{2} + C_{4}$ .

$y + \frac{1}{y} + C_{3} = 4 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.3.2

Simplify.

Tap for more steps...

Step 2.3.3.2.1

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

$y + \frac{1}{y} + C_{3} = \frac{4}{2} x^{2} + C_{4}$

Step 2.3.3.2.2

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

Tap for more steps...

Step 2.3.3.2.2.1

Factor $2$ out of $4$ .

$y + \frac{1}{y} + C_{3} = \frac{2 \cdot 2}{2} x^{2} + C_{4}$

Step 2.3.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

$y + \frac{1}{y} + C_{3} = \frac{2 \cdot 2}{2 (1)} x^{2} + C_{4}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$y + \frac{1}{y} + C_{3} = \frac{2 \cdot 2}{2 \cdot 1} x^{2} + C_{4}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$y + \frac{1}{y} + C_{3} = \frac{2}{1} x^{2} + C_{4}$

Step 2.3.3.2.2.2.4

Divide $2$ by $1$ .

$y + \frac{1}{y} + C_{3} = 2 x^{2} + C_{4}$

Step 2.4

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

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

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.

$1, y, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$y$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Multiply $y$ by $y$ .

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

Step 3.2.2.1.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.2.2.1.2.1

Cancel the common factor.

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

Step 3.2.2.1.2.2

Rewrite the expression.

$y^{2} + 1 = 2 x^{2} y + K y$

Step 3.3

Solve the equation.

Tap for more steps...

Step 3.3.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} y + K y = y^{2} + 1$

Step 3.3.2

Subtract $y^{2}$ from both sides of the equation.

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

Step 3.3.3

Subtract $1$ from both sides of the equation.

$2 x^{2} y + K y - y^{2} - 1 = 0$

Step 3.3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3.5

Substitute the values $a = - 1$ , $b = 2 x^{2} + K$ , and $c = - 1$ into the quadratic formula and solve for $y$ .

$\frac{- (2 x^{2} + K) \pm \sqrt{{(2 x^{2} + K)}^{2} - 4 \cdot (- 1 \cdot - 1)}}{2 \cdot - 1}$

Step 3.3.6

Simplify.

Tap for more steps...

Step 3.3.6.1

Simplify the numerator.

Tap for more steps...

Step 3.3.6.1.1

Apply the distributive property.

$y = \frac{- (2 x^{2}) - K \pm \sqrt{{(2 x^{2} + K)}^{2} - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.2

Multiply $2$ by $- 1$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{{(2 x^{2} + K)}^{2} - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.3

Rewrite ${(2 x^{2} + K)}^{2}$ as $(2 x^{2} + K) (2 x^{2} + K)$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{(2 x^{2} + K) (2 x^{2} + K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.4

Expand $(2 x^{2} + K) (2 x^{2} + K)$ using the FOIL Method.

Tap for more steps...

Step 3.3.6.1.4.1

Apply the distributive property.

$y = \frac{- 2 x^{2} - K \pm \sqrt{2 x^{2} (2 x^{2} + K) + K (2 x^{2} + K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.4.2

Apply the distributive property.

$y = \frac{- 2 x^{2} - K \pm \sqrt{2 x^{2} (2 x^{2}) + 2 x^{2} K + K (2 x^{2} + K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.4.3

Apply the distributive property.

$y = \frac{- 2 x^{2} - K \pm \sqrt{2 x^{2} (2 x^{2}) + 2 x^{2} K + K (2 x^{2}) + K (K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5

Simplify and combine like terms.

Tap for more steps...

Step 3.3.6.1.5.1

Simplify each term.

Tap for more steps...

Step 3.3.6.1.5.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{- 2 x^{2} - K \pm \sqrt{2 \cdot (2 x^{2} x^{2}) + 2 x^{2} K + K (2 x^{2}) + K (K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.3.6.1.5.1.2.1

Move $x^{2}$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{2 \cdot (2 (x^{2} x^{2})) + 2 x^{2} K + K (2 x^{2}) + K (K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{- 2 x^{2} - K \pm \sqrt{2 \cdot (2 x^{2 + 2}) + 2 x^{2} K + K (2 x^{2}) + K (K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.1.2.3

Add $2$ and $2$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{2 \cdot (2 x^{4}) + 2 x^{2} K + K (2 x^{2}) + K (K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.1.3

Multiply $2$ by $2$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{4 x^{4} + 2 x^{2} K + K (2 x^{2}) + K (K) - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 2 x^{2} - K \pm \sqrt{4 x^{4} + 2 x^{2} K + 2 K x^{2} + K \cdot K - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.1.5

Multiply $K$ by $K$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{4 x^{4} + 2 x^{2} K + 2 K x^{2} + K^{2} - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.2

Add $2 x^{2} K$ and $2 K x^{2}$ .

Tap for more steps...

Step 3.3.6.1.5.2.1

Move $x^{2}$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{4 x^{4} + 2 K x^{2} + 2 K x^{2} + K^{2} - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.5.2.2

Add $2 K x^{2}$ and $2 K x^{2}$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{4 x^{4} + 4 K x^{2} + K^{2} - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.6

Multiply $- 4 \cdot - 1 \cdot - 1$ .

Tap for more steps...

Step 3.3.6.1.6.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{4 x^{4} + 4 K x^{2} + K^{2} + 4 \cdot - 1}}{2 \cdot - 1}$

Step 3.3.6.1.6.2

Multiply $4$ by $- 1$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{4 x^{4} + 4 K x^{2} + K^{2} - 4}}{2 \cdot - 1}$

Step 3.3.6.1.7

Rewrite $4 x^{4} + 4 K x^{2} + K^{2} - 4$ in a factored form.

Tap for more steps...

Step 3.3.6.1.7.1

Factor using the perfect square rule.

Tap for more steps...

Step 3.3.6.1.7.1.1

Rewrite $4 x^{4}$ as ${(2 x^{2})}^{2}$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{{(2 x^{2})}^{2} + 4 K x^{2} + K^{2} - 4}}{2 \cdot - 1}$

Step 3.3.6.1.7.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 K x^{2} = 2 \cdot (2 x^{2}) \cdot K$

Step 3.3.6.1.7.1.3

Rewrite the polynomial.

$y = \frac{- 2 x^{2} - K \pm \sqrt{{(2 x^{2})}^{2} + 2 \cdot (2 x^{2}) \cdot K + K^{2} - 4}}{2 \cdot - 1}$

Step 3.3.6.1.7.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 x^{2}$ and $b = K$ .

$y = \frac{- 2 x^{2} - K \pm \sqrt{{(2 x^{2} + K)}^{2} - 4}}{2 \cdot - 1}$

Step 3.3.6.1.7.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.3.6.1.7.3

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

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

Step 3.3.6.2

Multiply $2$ by $- 1$ .

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

Step 3.3.6.3

Simplify $\frac{- 2 x^{2} - K \pm \sqrt{(2 x^{2} + K + 2) (2 x^{2} + K - 2)}}{- 2}$ .

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

Step 3.3.7

The final answer is the combination of both solutions.

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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