Calculus Examples

$\frac{d y}{d x} = \frac{6 x}{y - 2}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y - 2$ .

$(y - 2) \frac{d y}{d x} = (y - 2) \frac{6 x}{y - 2}$

Step 1.2

Cancel the common factor of $y - 2$ .

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Step 1.2.1

Cancel the common factor.

$(y - 2) \frac{d y}{d x} = (y - 2) \frac{6 x}{y - 2}$

Step 1.2.2

Rewrite the expression.

$(y - 2) \frac{d y}{d x} = 6 x$

Step 1.3

Rewrite the equation.

$(y - 2) d y = 6 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y - 2 d y = \int 6 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y d y + \int - 2 d y = \int 6 x d x$

Step 2.2.2

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

$\frac{1}{2} y^{2} + C_{1} + \int - 2 d y = \int 6 x d x$

Step 2.2.3

Apply the constant rule.

$\frac{1}{2} y^{2} + C_{1} - 2 y + C_{2} = \int 6 x d x$

Step 2.2.4

Simplify.

$\frac{1}{2} y^{2} - 2 y + C_{3} = \int 6 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{2} y^{2} - 2 y + C_{3} = 6 \int x d x$

Step 2.3.2

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

$\frac{1}{2} y^{2} - 2 y + C_{3} = 6 (\frac{1}{2} x^{2} + C_{4})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $6 (\frac{1}{2} x^{2} + C_{4})$ as $6 (\frac{1}{2}) x^{2} + C_{4}$ .

$\frac{1}{2} y^{2} - 2 y + C_{3} = 6 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $6$ and $\frac{1}{2}$ .

$\frac{1}{2} y^{2} - 2 y + C_{3} = \frac{6}{2} x^{2} + C_{4}$

Step 2.3.3.2.2

Cancel the common factor of $6$ and $2$ .

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Step 2.3.3.2.2.1

Factor $2$ out of $6$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $3$ by $1$ .

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

Step 2.4

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

$\frac{1}{2} y^{2} - 2 y = 3 x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

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

$\frac{y^{2}}{2} - 2 y = 3 x^{2} + K$

Step 3.2

Move all the expressions to the left side of the equation.

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Step 3.2.1

Subtract $3 x^{2}$ from both sides of the equation.

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

Step 3.2.2

Subtract $K$ from both sides of the equation.

$\frac{y^{2}}{2} - 2 y - 3 x^{2} - K = 0$

Step 3.3

Multiply through by the least common denominator $2$ , then simplify.

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Step 3.3.1

Apply the distributive property.

$2 (\frac{y^{2}}{2}) + 2 (- 2 y) + 2 (- 3 x^{2}) + 2 (- K) = 0$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$2 (\frac{y^{2}}{2}) + 2 (- 2 y) + 2 (- 3 x^{2}) + 2 (- K) = 0$

Step 3.3.2.1.2

Rewrite the expression.

$y^{2} + 2 (- 2 y) + 2 (- 3 x^{2}) + 2 (- K) = 0$

Step 3.3.2.2

Multiply $- 2$ by $2$ .

$y^{2} - 4 y + 2 (- 3 x^{2}) + 2 (- K) = 0$

Step 3.3.2.3

Multiply $- 3$ by $2$ .

$y^{2} - 4 y - 6 x^{2} + 2 (- K) = 0$

Step 3.3.2.4

Multiply $- 1$ by $2$ .

$y^{2} - 4 y - 6 x^{2} - 2 K = 0$

Step 3.3.3

Move $- 4 y$ .

$y^{2} - 6 x^{2} - 2 K - 4 y = 0$

Step 3.3.4

Reorder $y^{2}$ and $- 6 x^{2}$ .

$- 6 x^{2} + y^{2} - 2 K - 4 y = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 6 x^{2} - 2 K$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (- 6 x^{2} - 2 K))}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- 6 x^{2} - 2 K)$ .

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Step 3.6.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- 6 x^{2} - 2 K)}}{2 \cdot 1}$

Step 3.6.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- 6 x^{2} - 2 K)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- 6 x^{2} - 2 K))}}{2 \cdot 1}$

Step 3.6.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- 6 x^{2} - 2 K))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- 6 x^{2} - 2 K))}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 6 x^{2} - 2 K$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 6 x^{2} - 2 K)}}{2 \cdot 1}$

Step 3.6.1.3

Factor $2$ out of $- 4 - 6 x^{2} - 2 K$ .

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Step 3.6.1.3.1

Factor $2$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (2 (- 2) - 6 x^{2} - 2 K)}}{2 \cdot 1}$

Step 3.6.1.3.2

Factor $2$ out of $- 6 x^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 (2 (- 2) + 2 (- 3 x^{2}) - 2 K)}}{2 \cdot 1}$

Step 3.6.1.3.3

Factor $2$ out of $- 2 K$ .

$y = \frac{4 \pm \sqrt{- 4 (2 (- 2) + 2 (- 3 x^{2}) + 2 (- K))}}{2 \cdot 1}$

Step 3.6.1.3.4

Factor $2$ out of $2 (- 2) + 2 (- 3 x^{2})$ .

$y = \frac{4 \pm \sqrt{- 4 (2 (- 2 - 3 x^{2}) + 2 (- K))}}{2 \cdot 1}$

Step 3.6.1.3.5

Factor $2$ out of $2 (- 2 - 3 x^{2}) + 2 (- K)$ .

$y = \frac{4 \pm \sqrt{- 4 (2 (- 2 - 3 x^{2} - K))}}{2 \cdot 1}$

$y = \frac{4 \pm \sqrt{- 4 \cdot (2 (- 2 - 3 x^{2} - K))}}{2 \cdot 1}$

Step 3.6.1.4

Multiply $- 4$ by $2$ .

$y = \frac{4 \pm \sqrt{- 8 (- 2 - 3 x^{2} - K)}}{2 \cdot 1}$

Step 3.6.1.5

Rewrite $- 8 (- 2 - 3 x^{2} - K)$ as $2^{2} \cdot (- 2 (- 2 - 3 x^{2} - K))$ .

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Step 3.6.1.5.1

Factor $4$ out of $- 8$ .

$y = \frac{4 \pm \sqrt{4 (- 2) (- 2 - 3 x^{2} - K)}}{2 \cdot 1}$

Step 3.6.1.5.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 2 (- 2 - 3 x^{2} - K))}}{2 \cdot 1}$

Step 3.6.1.5.3

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 2 (- 2 - 3 x^{2} - K))}}{2 \cdot 1}$

Step 3.6.1.6

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 2 (- 2 - 3 x^{2} - K)}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 2 (- 2 - 3 x^{2} - K)}}{2}$

Step 3.6.3

Simplify $\frac{4 \pm 2 \sqrt{- 2 (- 2 - 3 x^{2} - K)}}{2}$ .

$y = 2 \pm \sqrt{- 2 (- 2 - 3 x^{2} - K)}$

Step 3.7

The final answer is the combination of both solutions.

$y = 2 + \sqrt{- 2 (- 2 - 3 x^{2} - K)}$

$y = 2 - \sqrt{- 2 (- 2 - 3 x^{2} - K)}$

$y = 2 + \sqrt{- 2 (- 2 - 3 x^{2} - K)}$

$y = 2 - \sqrt{- 2 (- 2 - 3 x^{2} - K)}$

Step 4

Simplify the constant of integration.

$y = 2 + \sqrt{- 2 (- 2 - 3 x^{2} + K)}$

$y = 2 - \sqrt{- 2 (- 2 - 3 x^{2} + K)}$