Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $x$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.2.2

Cancel the common factor of $\sin^{2} (\frac{y}{x})$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

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

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Cancel the common factor of $x$ .

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Step 1.3.1.1.1

Cancel the common factor.

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

Step 1.3.1.1.2

Rewrite the expression.

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Rewrite $\frac{1^{2}}{\sin^{2} (\frac{y}{x})}$ as ${(\frac{1}{\sin (\frac{y}{x})})}^{2}$ .

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

Step 1.3.1.4

Convert from $\frac{1}{\sin (\frac{y}{x})}$ to $\csc (\frac{y}{x})$ .

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

Step 1.3.1.5

Cancel the common factor of $\sin^{2} (\frac{y}{x})$ .

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Step 1.3.1.5.1

Cancel the common factor.

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

Step 1.3.1.5.2

Rewrite the expression.

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

Step 2

Let $V = \frac{y}{x}$ . Substitute $V$ for $\frac{y}{x}$ .

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

Step 3

Solve $V = \frac{y}{x}$ for $y$ .

$y = V x$

Step 4

Use the product rule to find the derivative of $y = V x$ with respect to $x$ .

$\frac{d y}{d x} = x \frac{d V}{d x} + V$

Step 5

Substitute $x \frac{d V}{d x} + V$ for $\frac{d y}{d x}$ .

$x \frac{d V}{d x} + V = \csc^{2} (V) + V$

Step 6

Solve the substituted differential equation.

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Step 6.1

Separate the variables.

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Step 6.1.1

Solve for $\frac{d V}{d x}$ .

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Step 6.1.1.1

Move all terms not containing $\frac{d V}{d x}$ to the right side of the equation.

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Step 6.1.1.1.1

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = \csc^{2} (V) + V - V$

Step 6.1.1.1.2

Combine the opposite terms in $\csc^{2} (V) + V - V$ .

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Step 6.1.1.1.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = \csc^{2} (V) + 0$

Step 6.1.1.1.2.2

Add $\csc^{2} (V)$ and $0$ .

$x \frac{d V}{d x} = \csc^{2} (V)$

Step 6.1.1.2

Divide each term in $x \frac{d V}{d x} = \csc^{2} (V)$ by $x$ and simplify.

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Step 6.1.1.2.1

Divide each term in $x \frac{d V}{d x} = \csc^{2} (V)$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{\csc^{2} (V)}{x}$

Step 6.1.1.2.2

Simplify the left side.

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Step 6.1.1.2.2.1

Cancel the common factor of $x$ .

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Step 6.1.1.2.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{\csc^{2} (V)}{x}$

Step 6.1.1.2.2.1.2

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

$\frac{d V}{d x} = \frac{\csc^{2} (V)}{x}$

Step 6.1.2

Multiply both sides by $\frac{1}{\csc^{2} (V)}$ .

$\frac{1}{\csc^{2} (V)} \frac{d V}{d x} = \frac{1}{\csc^{2} (V)} \cdot \frac{\csc^{2} (V)}{x}$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Combine.

$\frac{1}{\csc^{2} (V)} \frac{d V}{d x} = \frac{1 \csc^{2} (V)}{\csc^{2} (V) x}$

Step 6.1.3.2

Cancel the common factor of $\csc^{2} (V)$ .

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Step 6.1.3.2.1

Cancel the common factor.

$\frac{1}{\csc^{2} (V)} \frac{d V}{d x} = \frac{1 \csc^{2} (V)}{\csc^{2} (V) x}$

Step 6.1.3.2.2

Rewrite the expression.

$\frac{1}{\csc^{2} (V)} \frac{d V}{d x} = \frac{1}{x}$

Step 6.1.4

Rewrite the equation.

$\frac{1}{\csc^{2} (V)} d V = \frac{1}{x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{1}{\csc^{2} (V)} d V = \int \frac{1}{x} d x$

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Simplify.

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Step 6.2.2.1.1

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

$\int \frac{1^{2}}{\csc^{2} (V)} d V = \int \frac{1}{x} d x$

Step 6.2.2.1.2

Rewrite $\frac{1^{2}}{\csc^{2} (V)}$ as ${(\frac{1}{\csc (V)})}^{2}$ .

$\int {(\frac{1}{\csc (V)})}^{2} d V = \int \frac{1}{x} d x$

Step 6.2.2.1.3

Rewrite $\csc (V)$ in terms of sines and cosines.

$\int {(\frac{1}{\frac{1}{\sin (V)}})}^{2} d V = \int \frac{1}{x} d x$

Step 6.2.2.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (V)}$ .

$\int {(1 \sin (V))}^{2} d V = \int \frac{1}{x} d x$

Step 6.2.2.1.5

Multiply $\sin (V)$ by $1$ .

$\int \sin^{2} (V) d V = \int \frac{1}{x} d x$

Step 6.2.2.2

Use the half-angle formula to rewrite $\sin^{2} (V)$ as $\frac{1 - \cos (2 V)}{2}$ .

$\int \frac{1 - \cos (2 V)}{2} d V = \int \frac{1}{x} d x$

Step 6.2.2.3

Since $\frac{1}{2}$ is constant with respect to $V$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int 1 - \cos (2 V) d V = \int \frac{1}{x} d x$

Step 6.2.2.4

Split the single integral into multiple integrals.

$\frac{1}{2} (\int d V + \int - \cos (2 V) d V) = \int \frac{1}{x} d x$

Step 6.2.2.5

Apply the constant rule.

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

Step 6.2.2.6

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

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

Step 6.2.2.7

Let $u = 2 V$ . Then $d u = 2 d V$ , so $\frac{1}{2} d u = d V$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.2.7.1

Let $u = 2 V$ . Find $\frac{d u}{d V}$ .

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Step 6.2.2.7.1.1

Differentiate $2 V$ .

$\frac{d}{d V} [2 V]$

Step 6.2.2.7.1.2

Since $2$ is constant with respect to $V$ , the derivative of $2 V$ with respect to $V$ is $2 \frac{d}{d V} [V]$ .

$2 \frac{d}{d V} [V]$

Step 6.2.2.7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d V} [V^{n}]$ is $n V^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 6.2.2.7.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2.2.7.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{2} (V + C_{1} - \int \cos (u) \frac{1}{2} d u) = \int \frac{1}{x} d x$

Step 6.2.2.8

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{2} (V + C_{1} - \int \frac{\cos (u)}{2} d u) = \int \frac{1}{x} d x$

Step 6.2.2.9

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} (V + C_{1} - (\frac{1}{2} \int \cos (u) d u)) = \int \frac{1}{x} d x$

Step 6.2.2.10

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

$\frac{1}{2} (V + C_{1} - \frac{1}{2} (\sin (u) + C_{2})) = \int \frac{1}{x} d x$

Step 6.2.2.11

Simplify.

$\frac{1}{2} (V - \frac{1}{2} \sin (u)) + C_{3} = \int \frac{1}{x} d x$

Step 6.2.2.12

Replace all occurrences of $u$ with $2 V$ .

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

Step 6.2.2.13

Simplify.

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Step 6.2.2.13.1

Combine $\sin (2 V)$ and $\frac{1}{2}$ .

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

Step 6.2.2.13.2

Apply the distributive property.

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

Step 6.2.2.13.3

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

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

Step 6.2.2.13.4

Multiply $\frac{1}{2} (- \frac{\sin (2 V)}{2})$ .

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Step 6.2.2.13.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 V)}{2}$ .

$\frac{V}{2} - \frac{\sin (2 V)}{2 \cdot 2} + C_{3} = \int \frac{1}{x} d x$

Step 6.2.2.13.4.2

Multiply $2$ by $2$ .

$\frac{V}{2} - \frac{\sin (2 V)}{4} + C_{3} = \int \frac{1}{x} d x$

Step 6.2.2.14

Reorder terms.

$\frac{1}{2} V - \frac{1}{4} \sin (2 V) + C_{3} = \int \frac{1}{x} d x$

Step 6.2.3

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

$\frac{1}{2} V - \frac{1}{4} \sin (2 V) + C_{3} = \ln (| x |) + C_{4}$

Step 6.2.4

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

$\frac{1}{2} V - \frac{1}{4} \sin (2 V) = \ln (| x |) + C$

Step 7

Substitute $\frac{y}{x}$ for $V$ .

$\frac{1}{2} \cdot \frac{y}{x} - \frac{1}{4} \sin (2 \frac{y}{x}) = \ln (| x |) + C$

Step 8

Simplify the left side.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $\frac{1}{2}$ by $\frac{y}{x}$ .

$\frac{y}{2 x} - \frac{1}{4} \sin (2 \frac{y}{x}) = \ln (| x |) + C$

Step 8.1.2

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

$\frac{y}{2 x} - \frac{1}{4} \sin (\frac{2 y}{x}) = \ln (| x |) + C$

Step 8.1.3

Combine $\sin (\frac{2 y}{x})$ and $\frac{1}{4}$ .

$\frac{y}{2 x} - \frac{\sin (\frac{2 y}{x})}{4} = \ln (| x |) + C$