Calculus Examples

$y^{3} d x - x^{3} d y = 0$

Step 1

Subtract $y^{3} d x$ from both sides of the equation.

$- x^{3} d y = - y^{3} d x$

Step 2

Multiply both sides by $\frac{1}{x^{3} y^{3}}$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Cancel the common factor of $x^{3}$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{x^{3} y^{3}}$ into the numerator.

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

Step 3.2.2

Factor $x^{3}$ out of $x^{3} y^{3}$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Move the negative in front of the fraction.

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

Step 3.4

Rewrite using the commutative property of multiplication.

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

Step 3.5

Cancel the common factor of $y^{3}$ .

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Step 3.5.1

Move the leading negative in $- \frac{1}{x^{3} y^{3}}$ into the numerator.

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

Step 3.5.2

Factor $y^{3}$ out of $x^{3} y^{3}$ .

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

Apply basic rules of exponents.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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Step 4.2.2.2.1

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

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

Step 4.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 4.2.3

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

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

Step 4.2.4

Simplify the answer.

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Step 4.2.4.1

Simplify.

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Step 4.2.4.1.1

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

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

Step 4.2.4.1.2

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

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

Step 4.2.4.2

Simplify.

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

Step 4.2.4.3

Simplify.

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Step 4.2.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.4.3.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

Apply basic rules of exponents.

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 4.3.3

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

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

Step 4.3.4

Simplify the answer.

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Step 4.3.4.1

Simplify.

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Step 4.3.4.1.1

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

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

Step 4.3.4.1.2

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

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

Step 4.3.4.2

Simplify.

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

Step 4.3.4.3

Simplify.

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Step 4.3.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.4.3.2

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

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Find the LCD of the terms in the equation.

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Step 5.1.1

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

$2 y^{2}, 2 x^{2}, 1$

Step 5.1.2

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

Step 5.1.3

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 5.1.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 5.1.5

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

Not prime

Step 5.1.6

The LCM of $2, 2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 5.1.7

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

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

$y$ occurs $2$ times.

Step 5.1.8

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 5.1.9

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

$y \cdot y \cdot x \cdot x$

Step 5.1.10

Simplify $y \cdot y \cdot x \cdot x$ .

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Step 5.1.10.1

Multiply $y$ by $y$ .

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

Step 5.1.10.2

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

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Step 5.1.10.2.1

Move $x$ .

$y^{2} \cdot (x \cdot x)$

Step 5.1.10.2.2

Multiply $x$ by $x$ .

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

$y^{2} x^{2}$

Step 5.1.11

The LCM for $2 y^{2}, 2 x^{2}, 1$ is the numeric part $2$ multiplied by the variable part.

$2 y^{2} x^{2}$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Rewrite using the commutative property of multiplication.

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

Step 5.2.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.2.1

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

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

Step 5.2.2.2.2

Cancel the common factor.

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

Step 5.2.2.2.3

Rewrite the expression.

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

Step 5.2.2.3

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

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Step 5.2.2.3.1

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

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

Step 5.2.2.3.2

Cancel the common factor.

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

Step 5.2.2.3.3

Rewrite the expression.

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.2.3.1.2

Cancel the common factor of $2$ .

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Step 5.2.3.1.2.1

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

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.3

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

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Step 5.2.3.1.3.1

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

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

Step 5.2.3.1.3.2

Cancel the common factor.

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

Step 5.2.3.1.3.3

Rewrite the expression.

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

Step 5.2.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.3

Solve the equation.

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Step 5.3.1

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

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

Step 5.3.2

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

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Step 5.3.2.1

Multiply by $1$ .

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

Step 5.3.2.2

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

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

Step 5.3.2.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} (2 K x^{2})$ .

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

Step 5.3.3

Divide each term in $y^{2} (1 + 2 K x^{2}) = x^{2}$ by $1 + 2 K x^{2}$ and simplify.

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Step 5.3.3.1

Divide each term in $y^{2} (1 + 2 K x^{2}) = x^{2}$ by $1 + 2 K x^{2}$ .

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $1 + 2 K x^{2}$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $y^{2}$ by $1$ .

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

Step 5.3.4

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

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

Step 5.3.5

Simplify $\pm \sqrt{\frac{x^{2}}{1 + 2 K x^{2}}}$ .

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Step 5.3.5.1

Rewrite $\sqrt{\frac{x^{2}}{1 + 2 K x^{2}}}$ as $\frac{\sqrt{x^{2}}}{\sqrt{1 + 2 K x^{2}}}$ .

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

Step 5.3.5.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 5.3.5.3

Multiply $\frac{x}{\sqrt{1 + 2 K x^{2}}}$ by $\frac{\sqrt{1 + 2 K x^{2}}}{\sqrt{1 + 2 K x^{2}}}$ .

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

Step 5.3.5.4

Combine and simplify the denominator.

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Step 5.3.5.4.1

Multiply $\frac{x}{\sqrt{1 + 2 K x^{2}}}$ by $\frac{\sqrt{1 + 2 K x^{2}}}{\sqrt{1 + 2 K x^{2}}}$ .

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

Step 5.3.5.4.2

Raise $\sqrt{1 + 2 K x^{2}}$ to the power of $1$ .

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

Step 5.3.5.4.3

Raise $\sqrt{1 + 2 K x^{2}}$ to the power of $1$ .

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

Step 5.3.5.4.4

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

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

Step 5.3.5.4.5

Add $1$ and $1$ .

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

Step 5.3.5.4.6

Rewrite ${\sqrt{1 + 2 K x^{2}}}^{2}$ as $1 + 2 K x^{2}$ .

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Step 5.3.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + 2 K x^{2}}$ as ${(1 + 2 K x^{2})}^{\frac{1}{2}}$ .

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

Step 5.3.5.4.6.2

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

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

Step 5.3.5.4.6.3

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

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

Step 5.3.5.4.6.4

Cancel the common factor of $2$ .

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Step 5.3.5.4.6.4.1

Cancel the common factor.

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

Step 5.3.5.4.6.4.2

Rewrite the expression.

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

Step 5.3.5.4.6.5

Simplify.

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

Step 5.3.6

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

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Step 5.3.6.1

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

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

Step 5.3.6.2

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

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

Step 5.3.6.3

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

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

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

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

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

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

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

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

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

Step 6

Simplify the constant of integration.

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

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