Calculus Examples

$\frac{d y}{d x} = \frac{4 x}{3 y + 1}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $3 y + 1$ .

$(3 y + 1) \frac{d y}{d x} = (3 y + 1) \frac{4 x}{3 y + 1}$

Step 1.2

Cancel the common factor of $3 y + 1$ .

Tap for more steps...

Step 1.2.1

Cancel the common factor.

$(3 y + 1) \frac{d y}{d x} = (3 y + 1) \frac{4 x}{3 y + 1}$

Step 1.2.2

Rewrite the expression.

$(3 y + 1) \frac{d y}{d x} = 4 x$

Step 1.3

Rewrite the equation.

$(3 y + 1) d y = 4 x d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int 3 y + 1 d y = \int 4 x d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Split the single integral into multiple integrals.

$\int 3 y d y + \int d y = \int 4 x d x$

Step 2.2.2

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$3 \int y d y + \int d y = \int 4 x d x$

Step 2.2.3

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

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

Step 2.2.4

Apply the constant rule.

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

Step 2.2.5

Simplify.

Tap for more steps...

Step 2.2.5.1

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

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

Step 2.2.5.2

Simplify.

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

Step 2.2.5.3

Reorder terms.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

Tap for more steps...

Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

Tap for more steps...

Step 2.3.3.2.1

Combine $4$ and $\frac{1}{2}$ .

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

Step 2.3.3.2.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.3.3.2.2.1

Factor $2$ out of $4$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $2$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Combine $\frac{3}{2}$ and $y^{2}$ .

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Subtract $K$ from both sides of the equation.

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Simplify.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

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

Step 3.3.2.2

Multiply $- 2$ by $2$ .

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

Step 3.3.2.3

Multiply $- 1$ by $2$ .

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

Step 3.3.3

Move $2 y$ .

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

Step 3.3.4

Reorder $3 y^{2}$ and $- 4 x^{2}$ .

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

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 $a = 3$ , $b = 2$ , and $c = - 4 x^{2} - 2 K$ into the quadratic formula and solve for $y$ .

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

Step 3.6

Simplify.

Tap for more steps...

Step 3.6.1

Simplify the numerator.

Tap for more steps...

Step 3.6.1.1

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

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

Step 3.6.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 2 \pm \sqrt{4 - 12 \cdot (- 4 x^{2} - 2 K)}}{2 \cdot 3}$

Step 3.6.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 - 12 (- 4 x^{2}) - 12 (- 2 K)}}{2 \cdot 3}$

Step 3.6.1.4

Multiply $- 4$ by $- 12$ .

$y = \frac{- 2 \pm \sqrt{4 + 48 x^{2} - 12 (- 2 K)}}{2 \cdot 3}$

Step 3.6.1.5

Multiply $- 2$ by $- 12$ .

$y = \frac{- 2 \pm \sqrt{4 + 48 x^{2} + 24 K}}{2 \cdot 3}$

Step 3.6.1.6

Factor $4$ out of $4 + 48 x^{2} + 24 K$ .

Tap for more steps...

Step 3.6.1.6.1

Factor $4$ out of $4$ .

$y = \frac{- 2 \pm \sqrt{4 (1) + 48 x^{2} + 24 K}}{2 \cdot 3}$

Step 3.6.1.6.2

Factor $4$ out of $48 x^{2}$ .

$y = \frac{- 2 \pm \sqrt{4 (1) + 4 (12 x^{2}) + 24 K}}{2 \cdot 3}$

Step 3.6.1.6.3

Factor $4$ out of $24 K$ .

$y = \frac{- 2 \pm \sqrt{4 (1) + 4 (12 x^{2}) + 4 (6 K)}}{2 \cdot 3}$

Step 3.6.1.6.4

Factor $4$ out of $4 (1) + 4 (12 x^{2})$ .

$y = \frac{- 2 \pm \sqrt{4 (1 + 12 x^{2}) + 4 (6 K)}}{2 \cdot 3}$

Step 3.6.1.6.5

Factor $4$ out of $4 (1 + 12 x^{2}) + 4 (6 K)$ .

$y = \frac{- 2 \pm \sqrt{4 (1 + 12 x^{2} + 6 K)}}{2 \cdot 3}$

Step 3.6.1.7

Rewrite $4 (1 + 12 x^{2} + 6 K)$ as $2^{2} (1^{2} + 12 x^{2} + 6 K)$ .

Tap for more steps...

Step 3.6.1.7.1

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

$y = \frac{- 2 \pm \sqrt{2^{2} (1 + 12 x^{2} + 6 K)}}{2 \cdot 3}$

Step 3.6.1.7.2

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

$y = \frac{- 2 \pm \sqrt{2^{2} (1^{2} + 12 x^{2} + 6 K)}}{2 \cdot 3}$

Step 3.6.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{1^{2} + 12 x^{2} + 6 K}}{2 \cdot 3}$

Step 3.6.1.9

One to any power is one.

$y = \frac{- 2 \pm 2 \sqrt{1 + 12 x^{2} + 6 K}}{2 \cdot 3}$

Step 3.6.2

Multiply $2$ by $3$ .

$y = \frac{- 2 \pm 2 \sqrt{1 + 12 x^{2} + 6 K}}{6}$

Step 3.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{1 + 12 x^{2} + 6 K}}{6}$ .

$y = \frac{- 1 \pm \sqrt{1 + 12 x^{2} + 6 K}}{3}$

Step 3.7

The final answer is the combination of both solutions.

$y = - \frac{1 - \sqrt{1 + 12 x^{2} + 6 K}}{3}$

$y = - \frac{1 + \sqrt{12 x^{2} + 6 K + 1}}{3}$

$y = - \frac{1 - \sqrt{1 + 12 x^{2} + 6 K}}{3}$

$y = - \frac{1 + \sqrt{12 x^{2} + 6 K + 1}}{3}$

Step 4

Simplify the constant of integration.

$y = - \frac{1 - \sqrt{1 + 12 x^{2} + K}}{3}$

$y = - \frac{1 + \sqrt{12 x^{2} + K}}{3}$