Calculus Examples

$6 x (y d) x + (4 y + 9 y^{2}) d y = 0$

Step 1

Subtract $6 x y d x$ from both sides of the equation.

$(4 y + 9 y^{2}) d y = - 6 x y d x$

Step 2

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

$\frac{1}{y} (4 y + 9 y^{2}) d y = \frac{1}{y} (- 6 x y) d x$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{y}$ by $4 y + 9 y^{2}$ .

$\frac{4 y + 9 y^{2}}{y} d y = \frac{1}{y} (- 6 x y) d x$

Step 3.2

Factor $y$ out of $4 y + 9 y^{2}$ .

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Step 3.2.1

Factor $y$ out of $4 y$ .

$\frac{y \cdot 4 + 9 y^{2}}{y} d y = \frac{1}{y} (- 6 x y) d x$

Step 3.2.2

Factor $y$ out of $9 y^{2}$ .

$\frac{y \cdot 4 + y (9 y)}{y} d y = \frac{1}{y} (- 6 x y) d x$

Step 3.2.3

Factor $y$ out of $y \cdot 4 + y (9 y)$ .

$\frac{y (4 + 9 y)}{y} d y = \frac{1}{y} (- 6 x y) d x$

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

Cancel the common factor.

$\frac{y (4 + 9 y)}{y} d y = \frac{1}{y} (- 6 x y) d x$

Step 3.3.2

Divide $4 + 9 y$ by $1$ .

$(4 + 9 y) d y = \frac{1}{y} (- 6 x y) d x$

Step 3.4

Rewrite using the commutative property of multiplication.

$(4 + 9 y) d y = - 6 \frac{1}{y} (x y) d x$

Step 3.5

Combine $- 6$ and $\frac{1}{y}$ .

$(4 + 9 y) d y = \frac{- 6}{y} (x y) d x$

Step 3.6

Cancel the common factor of $y$ .

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Step 3.6.1

Factor $y$ out of $x y$ .

$(4 + 9 y) d y = \frac{- 6}{y} (y x) d x$

Step 3.6.2

Cancel the common factor.

$(4 + 9 y) d y = \frac{- 6}{y} (y x) d x$

Step 3.6.3

Rewrite the expression.

$(4 + 9 y) d y = - 6 x d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int 4 + 9 y d y = \int - 6 x d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Split the single integral into multiple integrals.

$\int 4 d y + \int 9 y d y = \int - 6 x d x$

Step 4.2.2

Apply the constant rule.

$4 y + C_{1} + \int 9 y d y = \int - 6 x d x$

Step 4.2.3

Since $9$ is constant with respect to $y$ , move $9$ out of the integral.

$4 y + C_{1} + 9 \int y d y = \int - 6 x d x$

Step 4.2.4

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

$4 y + C_{1} + 9 (\frac{1}{2} y^{2} + C_{2}) = \int - 6 x d x$

Step 4.2.5

Simplify.

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Step 4.2.5.1

Simplify.

$4 y + 9 (\frac{1}{2}) y^{2} + C_{3} = \int - 6 x d x$

Step 4.2.5.2

Combine $9$ and $\frac{1}{2}$ .

$4 y + \frac{9}{2} y^{2} + C_{3} = \int - 6 x d x$

Step 4.2.6

Reorder terms.

$\frac{9}{2} y^{2} + 4 y + C_{3} = \int - 6 x d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

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

$\frac{9}{2} y^{2} + 4 y + C_{3} = - 6 \int x d x$

Step 4.3.2

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

$\frac{9}{2} y^{2} + 4 y + C_{3} = - 6 (\frac{1}{2} x^{2} + C_{4})$

Step 4.3.3

Simplify the answer.

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Step 4.3.3.1

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

$\frac{9}{2} y^{2} + 4 y + C_{3} = - 6 (\frac{1}{2}) x^{2} + C_{4}$

Step 4.3.3.2

Simplify.

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Step 4.3.3.2.1

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

$\frac{9}{2} y^{2} + 4 y + C_{3} = \frac{- 6}{2} x^{2} + C_{4}$

Step 4.3.3.2.2

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

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Step 4.3.3.2.2.1

Factor $2$ out of $- 6$ .

$\frac{9}{2} y^{2} + 4 y + C_{3} = \frac{2 \cdot - 3}{2} x^{2} + C_{4}$

Step 4.3.3.2.2.2

Cancel the common factors.

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Step 4.3.3.2.2.2.1

Factor $2$ out of $2$ .

$\frac{9}{2} y^{2} + 4 y + C_{3} = \frac{2 \cdot - 3}{2 (1)} x^{2} + C_{4}$

Step 4.3.3.2.2.2.2

Cancel the common factor.

$\frac{9}{2} y^{2} + 4 y + C_{3} = \frac{2 \cdot - 3}{2 \cdot 1} x^{2} + C_{4}$

Step 4.3.3.2.2.2.3

Rewrite the expression.

$\frac{9}{2} y^{2} + 4 y + C_{3} = \frac{- 3}{1} x^{2} + C_{4}$

Step 4.3.3.2.2.2.4

Divide $- 3$ by $1$ .

$\frac{9}{2} y^{2} + 4 y + C_{3} = - 3 x^{2} + C_{4}$

Step 4.4

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

$\frac{9}{2} y^{2} + 4 y = - 3 x^{2} + K$

Step 5

Solve for $y$ .

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Step 5.1

Combine $\frac{9}{2}$ and $y^{2}$ .

$\frac{9 y^{2}}{2} + 4 y = - 3 x^{2} + K$

Step 5.2

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

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Step 5.2.1

Add $3 x^{2}$ to both sides of the equation.

$\frac{9 y^{2}}{2} + 4 y + 3 x^{2} = K$

Step 5.2.2

Subtract $K$ from both sides of the equation.

$\frac{9 y^{2}}{2} + 4 y + 3 x^{2} - K = 0$

Step 5.3

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

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Step 5.3.1

Apply the distributive property.

$2 (\frac{9 y^{2}}{2}) + 2 (4 y) + 2 (3 x^{2}) + 2 (- K) = 0$

Step 5.3.2

Simplify.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$2 (\frac{9 y^{2}}{2}) + 2 (4 y) + 2 (3 x^{2}) + 2 (- K) = 0$

Step 5.3.2.1.2

Rewrite the expression.

$9 y^{2} + 2 (4 y) + 2 (3 x^{2}) + 2 (- K) = 0$

Step 5.3.2.2

Multiply $4$ by $2$ .

$9 y^{2} + 8 y + 2 (3 x^{2}) + 2 (- K) = 0$

Step 5.3.2.3

Multiply $3$ by $2$ .

$9 y^{2} + 8 y + 6 x^{2} + 2 (- K) = 0$

Step 5.3.2.4

Multiply $- 1$ by $2$ .

$9 y^{2} + 8 y + 6 x^{2} - 2 K = 0$

Step 5.3.3

Move $8 y$ .

$9 y^{2} + 6 x^{2} - 2 K + 8 y = 0$

Step 5.3.4

Reorder $9 y^{2}$ and $6 x^{2}$ .

$6 x^{2} + 9 y^{2} - 2 K + 8 y = 0$

Step 5.4

Use the quadratic formula to find the solutions.

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

Step 5.5

Substitute the values $a = 9$ , $b = 8$ , and $c = 6 x^{2} - 2 K$ into the quadratic formula and solve for $y$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (9 \cdot (6 x^{2} - 2 K))}}{2 \cdot 9}$

Step 5.6

Simplify.

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

Raise $8$ to the power of $2$ .

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot 9 \cdot (6 x^{2} - 2 K)}}{2 \cdot 9}$

Step 5.6.1.2

Multiply $- 4$ by $9$ .

$y = \frac{- 8 \pm \sqrt{64 - 36 \cdot (6 x^{2} - 2 K)}}{2 \cdot 9}$

Step 5.6.1.3

Apply the distributive property.

$y = \frac{- 8 \pm \sqrt{64 - 36 (6 x^{2}) - 36 (- 2 K)}}{2 \cdot 9}$

Step 5.6.1.4

Multiply $6$ by $- 36$ .

$y = \frac{- 8 \pm \sqrt{64 - 216 x^{2} - 36 (- 2 K)}}{2 \cdot 9}$

Step 5.6.1.5

Multiply $- 2$ by $- 36$ .

$y = \frac{- 8 \pm \sqrt{64 - 216 x^{2} + 72 K}}{2 \cdot 9}$

Step 5.6.1.6

Factor $8$ out of $64 - 216 x^{2} + 72 K$ .

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Step 5.6.1.6.1

Factor $8$ out of $64$ .

$y = \frac{- 8 \pm \sqrt{8 (8) - 216 x^{2} + 72 K}}{2 \cdot 9}$

Step 5.6.1.6.2

Factor $8$ out of $- 216 x^{2}$ .

$y = \frac{- 8 \pm \sqrt{8 (8) + 8 (- 27 x^{2}) + 72 K}}{2 \cdot 9}$

Step 5.6.1.6.3

Factor $8$ out of $72 K$ .

$y = \frac{- 8 \pm \sqrt{8 (8) + 8 (- 27 x^{2}) + 8 (9 K)}}{2 \cdot 9}$

Step 5.6.1.6.4

Factor $8$ out of $8 (8) + 8 (- 27 x^{2})$ .

$y = \frac{- 8 \pm \sqrt{8 (8 - 27 x^{2}) + 8 (9 K)}}{2 \cdot 9}$

Step 5.6.1.6.5

Factor $8$ out of $8 (8 - 27 x^{2}) + 8 (9 K)$ .

$y = \frac{- 8 \pm \sqrt{8 (8 - 27 x^{2} + 9 K)}}{2 \cdot 9}$

Step 5.6.1.7

Rewrite $8 (8 - 27 x^{2} + 9 K)$ as $2^{2} \cdot (2 (8 - 27 x^{2} + 3^{2} K))$ .

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Step 5.6.1.7.1

Factor $4$ out of $8$ .

$y = \frac{- 8 \pm \sqrt{4 (2) (8 - 27 x^{2} + 9 K)}}{2 \cdot 9}$

Step 5.6.1.7.2

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

$y = \frac{- 8 \pm \sqrt{2^{2} \cdot (2 (8 - 27 x^{2} + 9 K))}}{2 \cdot 9}$

Step 5.6.1.7.3

Rewrite $9$ as $3^{2}$ .

$y = \frac{- 8 \pm \sqrt{2^{2} \cdot (2 (8 - 27 x^{2} + 3^{2} K))}}{2 \cdot 9}$

Step 5.6.1.7.4

Add parentheses.

$y = \frac{- 8 \pm \sqrt{2^{2} \cdot (2 (8 - 27 x^{2} + 3^{2} K))}}{2 \cdot 9}$

Step 5.6.1.8

Pull terms out from under the radical.

$y = \frac{- 8 \pm 2 \sqrt{2 (8 - 27 x^{2} + 3^{2} K)}}{2 \cdot 9}$

Step 5.6.1.9

Raise $3$ to the power of $2$ .

$y = \frac{- 8 \pm 2 \sqrt{2 (8 - 27 x^{2} + 9 K)}}{2 \cdot 9}$

Step 5.6.2

Multiply $2$ by $9$ .

$y = \frac{- 8 \pm 2 \sqrt{2 (8 - 27 x^{2} + 9 K)}}{18}$

Step 5.6.3

Simplify $\frac{- 8 \pm 2 \sqrt{2 (8 - 27 x^{2} + 9 K)}}{18}$ .

$y = \frac{- 4 \pm \sqrt{2 (8 - 27 x^{2} + 9 K)}}{9}$

Step 5.7

The final answer is the combination of both solutions.

$y = - \frac{4 - \sqrt{2 (8 - 27 x^{2} + 9 K)}}{9}$

$y = - \frac{\sqrt{2 (- 27 x^{2} + 9 K + 8)} + 4}{9}$

$y = - \frac{4 - \sqrt{2 (8 - 27 x^{2} + 9 K)}}{9}$

$y = - \frac{\sqrt{2 (- 27 x^{2} + 9 K + 8)} + 4}{9}$

Step 6

Simplify the constant of integration.

$y = - \frac{4 - \sqrt{2 (8 - 27 x^{2} + K)}}{9}$

$y = - \frac{\sqrt{2 (- 27 x^{2} + K)} + 4}{9}$