Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $1 + 2 y$ .

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

Step 1.2

Cancel the common factor of $1 + 2 y$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

Apply the constant rule.

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

Step 2.2.3

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

Simplify.

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

Step 2.2.5.2

Simplify.

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Step 2.2.5.2.1

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

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

Step 2.2.5.2.2

Cancel the common factor of $2$ .

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Step 2.2.5.2.2.1

Cancel the common factor.

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

Step 2.2.5.2.2.2

Rewrite the expression.

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

Step 2.2.5.2.3

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

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

Step 2.2.6

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Subtract $K$ from both sides of the equation.

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

Step 3.2

Use the quadratic formula to find the solutions.

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

One to any power is one.

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

Step 3.4.1.2

Multiply $- 4$ by $1$ .

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

Step 3.4.1.3

Apply the distributive property.

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

Step 3.4.1.4

Multiply $- 1$ by $- 4$ .

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

Step 3.4.1.5

Multiply $- 1$ by $- 4$ .

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

Step 3.4.2

Multiply $2$ by $1$ .

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

Step 3.5

The final answer is the combination of both solutions.

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

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

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

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

Step 4

Simplify the constant of integration.

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

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