Calculus Examples

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

Step 1

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

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

Step 2

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

$y = V x$

Step 3

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 4

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

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

Step 5

Solve the substituted differential equation.

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Step 5.1

Separate the variables.

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Step 5.1.1

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

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Step 5.1.1.1

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

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Step 5.1.1.1.1

Subtract $V$ from both sides of the equation.

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

Step 5.1.1.1.2

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

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Step 5.1.1.1.2.1

Subtract $V$ from $V$ .

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

Step 5.1.1.1.2.2

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

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

Step 5.1.1.2

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

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Step 5.1.1.2.1

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

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

Step 5.1.1.2.2

Simplify the left side.

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Step 5.1.1.2.2.1

Cancel the common factor of $x$ .

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Step 5.1.1.2.2.1.1

Cancel the common factor.

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

Step 5.1.1.2.2.1.2

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

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

Step 5.1.2

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

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

Step 5.1.3

Simplify.

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Step 5.1.3.1

Combine.

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

Step 5.1.3.2

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

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Step 5.1.3.2.1

Cancel the common factor.

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

Step 5.1.3.2.2

Rewrite the expression.

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

Step 5.1.4

Rewrite the equation.

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

Step 5.2

Integrate both sides.

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Step 5.2.1

Set up an integral on each side.

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

Step 5.2.2

Integrate the left side.

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Step 5.2.2.1

Convert from $\frac{1}{\cos^{2} (V)}$ to $\sec^{2} (V)$ .

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

Step 5.2.2.2

Since the derivative of $\tan (V)$ is $\sec^{2} (V)$ , the integral of $\sec^{2} (V)$ is $\tan (V)$ .

$\tan (V) + C_{1} = \int \frac{1}{x} d x$

Step 5.2.3

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

$\tan (V) + C_{1} = \ln (| x |) + C_{2}$

Step 5.2.4

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

$\tan (V) = \ln (| x |) + C$

Step 5.3

Take the inverse tangent of both sides of the equation to extract $V$ from inside the tangent.

$V = \arctan (\ln (| x |) + C)$

Step 6

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

$\frac{y}{x} = \arctan (\ln (| x |) + C)$

Step 7

Solve $\frac{y}{x} = \arctan (\ln (| x |) + C)$ for $y$ .

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Step 7.1

Multiply both sides by $x$ .

$\frac{y}{x} x = \arctan (\ln (| x |) + C) x$

Step 7.2

Simplify.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $x$ .

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Step 7.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = \arctan (\ln (| x |) + C) x$

Step 7.2.1.1.2

Rewrite the expression.

$y = \arctan (\ln (| x |) + C) x$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Reorder factors in $\arctan (\ln (| x |) + C) x$ .

$y = x \arctan (\ln (| x |) + C)$