Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y + 1$ .

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

Step 1.2

Cancel the common factor of $y + 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$(y + 1) d y = (x - 2) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y + 1 d y = \int x - 2 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Apply the constant rule.

$\frac{1}{2} y^{2} + y + C_{3} = \frac{1}{2} x^{2} + C_{4} - 2 x + C_{5}$

Step 2.3.4

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Add $2 x$ to both sides of the equation.

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

Step 3.3.3

Subtract $K$ from both sides of the equation.

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

Step 3.4

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

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Simplify.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

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

Step 3.4.2.2

Cancel the common factor of $2$ .

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

Cancel the common factor.

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

Step 3.4.2.2.3

Rewrite the expression.

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

Step 3.4.2.3

Multiply $2$ by $2$ .

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

Step 3.4.2.4

Multiply $- 1$ by $2$ .

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

Step 3.4.3

Move $2 y$ .

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

Step 3.4.4

Move $4 x$ .

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

Step 3.4.5

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

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

Step 3.5

Use the quadratic formula to find the solutions.

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

Step 3.6

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

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

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $2$ to the power of $2$ .

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

Step 3.7.1.2

Multiply $- 4$ by $1$ .

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

Step 3.7.1.3

Apply the distributive property.

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

Step 3.7.1.4

Simplify.

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Step 3.7.1.4.1

Multiply $- 1$ by $- 4$ .

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

Step 3.7.1.4.2

Multiply $- 2$ by $- 4$ .

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

Step 3.7.1.4.3

Multiply $4$ by $- 4$ .

$y = \frac{- 2 \pm \sqrt{4 + 4 x^{2} + 8 K - 16 x}}{2 \cdot 1}$

Step 3.7.1.5

Factor $4$ out of $4 + 4 x^{2} + 8 K - 16 x$ .

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Step 3.7.1.5.1

Factor $4$ out of $4$ .

$y = \frac{- 2 \pm \sqrt{4 \cdot 1 + 4 x^{2} + 8 K - 16 x}}{2 \cdot 1}$

Step 3.7.1.5.2

Factor $4$ out of $8 K$ .

$y = \frac{- 2 \pm \sqrt{4 \cdot 1 + 4 x^{2} + 4 (2 K) - 16 x}}{2 \cdot 1}$

Step 3.7.1.5.3

Factor $4$ out of $- 16 x$ .

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

Step 3.7.1.5.4

Factor $4$ out of $4 \cdot 1 + 4 x^{2}$ .

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

Step 3.7.1.5.5

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

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

Step 3.7.1.5.6

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

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

Step 3.7.1.6

Rewrite $4 (1 + x^{2} + 2 K - 4 x)$ as $2^{2} (1^{2} + x^{2} + 2 K - 4 x)$ .

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Step 3.7.1.6.1

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

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

Step 3.7.1.6.2

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

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

Step 3.7.1.7

Pull terms out from under the radical.

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

Step 3.7.1.8

One to any power is one.

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

Step 3.7.2

Multiply $2$ by $1$ .

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

Step 3.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{1 + x^{2} + 2 K - 4 x}}{2}$ .

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

Step 3.8

The final answer is the combination of both solutions.

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

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

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

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

Step 4

Simplify the constant of integration.

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

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