Calculus Examples

$\frac{d y}{d x} = \frac{\sin (x)}{y + 2}$ ; with $y (0) = 2$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y + 2$ .

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

Step 1.2

Cancel the common factor of $y + 2$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$(y + 2) d y = \sin (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y + 2 d y = \int \sin (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y d y + \int 2 d y = \int \sin (x) d x$

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $2$ for $y$ in $\frac{1}{2} y^{2} + 2 y = - \cos (x) + K$ .

$\frac{1}{2} \cdot 2^{2} + 2 \cdot 2 = - \cos (0) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \cos (0) + K = \frac{1}{2} \cdot 2^{2} + 2 \cdot 2$ .

$- \cos (0) + K = \frac{1}{2} \cdot 2^{2} + 2 \cdot 2$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

The exact value of $\cos (0)$ is $1$ .

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

Step 4.2.1.2

Multiply $- 1$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify $\frac{1}{2} \cdot 2^{2} + 2 \cdot 2$ .

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Step 4.3.1.1

Simplify each term.

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Step 4.3.1.1.1

Cancel the common factor of $2$ .

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Step 4.3.1.1.1.1

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

$- 1 + K = \frac{1}{2} \cdot (2 \cdot 2) + 2 \cdot 2$

Step 4.3.1.1.1.2

Cancel the common factor.

$- 1 + K = \frac{1}{2} \cdot (2 \cdot 2) + 2 \cdot 2$

Step 4.3.1.1.1.3

Rewrite the expression.

$- 1 + K = 2 + 2 \cdot 2$

Step 4.3.1.1.2

Multiply $2$ by $2$ .

$- 1 + K = 2 + 4$

Step 4.3.1.2

Add $2$ and $4$ .

$- 1 + K = 6$

Step 4.4

Move all terms not containing $K$ to the right side of the equation.

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Step 4.4.1

Add $1$ to both sides of the equation.

$K = 6 + 1$

Step 4.4.2

Add $6$ and $1$ .

$K = 7$

Step 5

Substitute $7$ for $K$ in $\frac{1}{2} y^{2} + 2 y = - \cos (x) + K$ and simplify.

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Step 5.1

Substitute $7$ for $K$ .

$\frac{1}{2} y^{2} + 2 y = - \cos (x) + 7$

Step 5.2

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

$\frac{y^{2}}{2} + 2 y = - \cos (x) + 7$