Calculus Examples

$\frac{d y}{d x} = \frac{\cos (x)}{3 y - y^{2}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $3 y - y^{2}$ .

$(3 y - y^{2}) \frac{d y}{d x} = (3 y - y^{2}) \frac{\cos (x)}{3 y - y^{2}}$

Step 1.2

Simplify.

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Step 1.2.1

Factor $y$ out of $3 y - y^{2}$ .

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Step 1.2.1.1

Factor $y$ out of $3 y$ .

$(3 y - y^{2}) \frac{d y}{d x} = (3 y - y^{2}) \frac{\cos (x)}{y \cdot 3 - y^{2}}$

Step 1.2.1.2

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

$(3 y - y^{2}) \frac{d y}{d x} = (3 y - y^{2}) \frac{\cos (x)}{y \cdot 3 + y (- y)}$

Step 1.2.1.3

Factor $y$ out of $y \cdot 3 + y (- y)$ .

$(3 y - y^{2}) \frac{d y}{d x} = (3 y - y^{2}) \frac{\cos (x)}{y (3 - y)}$

Step 1.2.2

Apply the distributive property.

$(3 y - y^{2}) \frac{d y}{d x} = 3 y \frac{\cos (x)}{y (3 - y)} - y^{2} \frac{\cos (x)}{y (3 - y)}$

Step 1.2.3

Cancel the common factor of $y$ .

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Step 1.2.3.1

Factor $y$ out of $3 y$ .

$(3 y - y^{2}) \frac{d y}{d x} = y \cdot 3 \frac{\cos (x)}{y (3 - y)} - y^{2} \frac{\cos (x)}{y (3 - y)}$

Step 1.2.3.2

Cancel the common factor.

$(3 y - y^{2}) \frac{d y}{d x} = y \cdot 3 \frac{\cos (x)}{y (3 - y)} - y^{2} \frac{\cos (x)}{y (3 - y)}$

Step 1.2.3.3

Rewrite the expression.

$(3 y - y^{2}) \frac{d y}{d x} = 3 \frac{\cos (x)}{3 - y} - y^{2} \frac{\cos (x)}{y (3 - y)}$

Step 1.2.4

Combine $3$ and $\frac{\cos (x)}{3 - y}$ .

$(3 y - y^{2}) \frac{d y}{d x} = \frac{3 \cos (x)}{3 - y} - y^{2} \frac{\cos (x)}{y (3 - y)}$

Step 1.2.5

Cancel the common factor of $y$ .

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Step 1.2.5.1

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

$(3 y - y^{2}) \frac{d y}{d x} = \frac{3 \cos (x)}{3 - y} + y (- y) \frac{\cos (x)}{y (3 - y)}$

Step 1.2.5.2

Cancel the common factor.

$(3 y - y^{2}) \frac{d y}{d x} = \frac{3 \cos (x)}{3 - y} + y (- y) \frac{\cos (x)}{y (3 - y)}$

Step 1.2.5.3

Rewrite the expression.

$(3 y - y^{2}) \frac{d y}{d x} = \frac{3 \cos (x)}{3 - y} - y \frac{\cos (x)}{3 - y}$

Step 1.2.6

Combine $\frac{\cos (x)}{3 - y}$ and $y$ .

$(3 y - y^{2}) \frac{d y}{d x} = \frac{3 \cos (x)}{3 - y} - \frac{\cos (x) y}{3 - y}$

Step 1.2.7

Combine the numerators over the common denominator.

$(3 y - y^{2}) \frac{d y}{d x} = \frac{3 \cos (x) - \cos (x) y}{3 - y}$

Step 1.2.8

Factor $\cos (x)$ out of $3 \cos (x) - \cos (x) y$ .

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Step 1.2.8.1

Factor $\cos (x)$ out of $3 \cos (x)$ .

$(3 y - y^{2}) \frac{d y}{d x} = \frac{\cos (x) \cdot 3 - \cos (x) y}{3 - y}$

Step 1.2.8.2

Factor $\cos (x)$ out of $- \cos (x) y$ .

$(3 y - y^{2}) \frac{d y}{d x} = \frac{\cos (x) \cdot 3 + \cos (x) (- y)}{3 - y}$

Step 1.2.8.3

Factor $\cos (x)$ out of $\cos (x) \cdot 3 + \cos (x) (- y)$ .

$(3 y - y^{2}) \frac{d y}{d x} = \frac{\cos (x) (3 - y)}{3 - y}$

Step 1.2.9

Cancel the common factor of $3 - y$ .

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Step 1.2.9.1

Cancel the common factor.

$(3 y - y^{2}) \frac{d y}{d x} = \frac{\cos (x) (3 - y)}{3 - y}$

Step 1.2.9.2

Divide $\cos (x)$ by $1$ .

$(3 y - y^{2}) \frac{d y}{d x} = \cos (x)$

Step 1.3

Rewrite the equation.

$(3 y - y^{2}) d y = \cos (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 3 y - y^{2} d y = \int \cos (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int 3 y d y + \int - y^{2} d y = \int \cos (x) d x$

Step 2.2.2

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

$3 \int y d y + \int - y^{2} d y = \int \cos (x) d x$

Step 2.2.3

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

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

Step 2.2.4

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

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

Step 2.2.5

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

Simplify.

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

Step 2.2.6.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 2.3

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

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

Step 2.4

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

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