Calculus Examples

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

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Divide each term in $x \frac{d y}{d x} + 2 y = \frac{\sin (x)}{x}$ by $x$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.3

Factor $y$ out of $\frac{2 y}{x}$ .

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

Step 1.4

Reorder $y$ and $\frac{2}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{2}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{2}{x} d x}$

Step 2.2

Integrate $\frac{2}{x}$ .

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Step 2.2.1

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

$e^{2 \int \frac{1}{x} d x}$

Step 2.2.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$e^{2 (\ln (| x |) + C)}$

Step 2.2.3

Simplify.

$e^{2 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{2 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{2})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 3

Multiply each term by the integrating factor $x^{2}$ .

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Step 3.1

Multiply each term by $x^{2}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{2}{x}$ and $y$ .

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Combine $x$ and $\frac{\sin (x)}{x}$ .

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

Step 3.5

Cancel the common factor of $x$ .

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Step 3.5.1

Cancel the common factor.

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

Step 3.5.2

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

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

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [x^{2} y] = \sin (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int \sin (x) d x$

Step 6

Integrate the left side.

$x^{2} y = \int \sin (x) d x$

Step 7

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

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

Step 8

Divide each term in $x^{2} y = - \cos (x) + C$ by $x^{2}$ and simplify.

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Step 8.1

Divide each term in $x^{2} y = - \cos (x) + C$ by $x^{2}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x^{2}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

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

Step 9

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

$1 = - \frac{\cos (2)}{2^{2}} + \frac{C}{2^{2}}$

Step 10

Solve for $C$ .

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Step 10.1

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

$- \frac{\cos (2)}{2^{2}} + \frac{C}{2^{2}} = 1$

Step 10.2

Simplify the left side.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Evaluate $\cos (2)$ .

$- \frac{0.99939082}{2^{2}} + \frac{C}{2^{2}} = 1$

Step 10.2.1.2

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

$- \frac{0.99939082}{4} + \frac{C}{2^{2}} = 1$

Step 10.2.1.3

Divide $0.99939082$ by $4$ .

$- 1 \cdot 0.2498477 + \frac{C}{2^{2}} = 1$

Step 10.2.1.4

Multiply $- 1$ by $0.2498477$ .

$- 0.2498477 + \frac{C}{2^{2}} = 1$

Step 10.2.1.5

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

$- 0.2498477 + \frac{C}{4} = 1$

Step 10.3

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

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Step 10.3.1

Add $0.2498477$ to both sides of the equation.

$\frac{C}{4} = 1 + 0.2498477$

Step 10.3.2

Add $1$ and $0.2498477$ .

$\frac{C}{4} = 1.2498477$

Step 10.4

Multiply both sides of the equation by $4$ .

$4 \frac{C}{4} = 4 \cdot 1.2498477$

Step 10.5

Simplify both sides of the equation.

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Step 10.5.1

Simplify the left side.

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Step 10.5.1.1

Cancel the common factor of $4$ .

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Step 10.5.1.1.1

Cancel the common factor.

$4 \frac{C}{4} = 4 \cdot 1.2498477$

Step 10.5.1.1.2

Rewrite the expression.

$C = 4 \cdot 1.2498477$

Step 10.5.2

Simplify the right side.

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Step 10.5.2.1

Multiply $4$ by $1.2498477$ .

$C = 4.99939082$

Step 11

Substitute $4.99939082$ for $C$ in $y = - \frac{\cos (x)}{x^{2}} + \frac{C}{x^{2}}$ and simplify.

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Step 11.1

Substitute $4.99939082$ for $C$ .

$y = - \frac{\cos (x)}{x^{2}} + \frac{4.99939082}{x^{2}}$