Calculus Examples

$x^{2} \frac{d y}{d x} = y^{2} + x y + 4 x^{2}$

Step 1

Rewrite the differential equation as a function of $\frac{y}{x}$ .

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Step 1.1

Divide each term in $x^{2} \frac{d y}{d x} = y^{2} + x y + 4 x^{2}$ by $x^{2}$ and simplify.

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Step 1.1.1

Divide each term in $x^{2} \frac{d y}{d x} = y^{2} + x y + 4 x^{2}$ by $x^{2}$ .

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} = \frac{y^{2}}{x^{2}} + \frac{x y}{x^{2}} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} = \frac{y^{2}}{x^{2}} + \frac{x y}{x^{2}} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.2.1.2

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

$\frac{d y}{d x} = \frac{y^{2}}{x^{2}} + \frac{x y}{x^{2}} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

Factor $x$ out of $x y$ .

$\frac{d y}{d x} = \frac{y^{2}}{x^{2}} + \frac{x (y)}{x^{2}} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

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

$\frac{d y}{d x} = \frac{y^{2}}{x^{2}} + \frac{x (y)}{x \cdot x} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.3.1.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{y^{2}}{x^{2}} + \frac{x y}{x \cdot x} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.3.1.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{y^{2}}{x^{2}} + \frac{y}{x} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{y^{2}}{x^{2}} + \frac{y}{x} + \frac{4 x^{2}}{x^{2}}$

Step 1.1.3.1.2.2

Divide $4$ by $1$ .

$\frac{d y}{d x} = \frac{y^{2}}{x^{2}} + \frac{y}{x} + 4$

Step 1.2

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

$\frac{d y}{d x} = {(\frac{y}{x})}^{2} + \frac{y}{x} + 4$

Step 2

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

$\frac{d y}{d x} = V^{2} + V + 4$

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 $\frac{d y}{d x}$ .

$x \frac{d V}{d x} + V = V^{2} + V + 4$

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.

$x \frac{d V}{d x} = V^{2} + V + 4 - V$

Step 6.1.1.1.2

Combine the opposite terms in $V^{2} + V + 4 - V$ .

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Step 6.1.1.1.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = V^{2} + 0 + 4$

Step 6.1.1.1.2.2

Add $V^{2}$ and $0$ .

$x \frac{d V}{d x} = V^{2} + 4$

Step 6.1.1.2

Divide each term in $x \frac{d V}{d x} = V^{2} + 4$ by $x$ and simplify.

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Step 6.1.1.2.1

Divide each term in $x \frac{d V}{d x} = V^{2} + 4$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{V^{2}}{x} + \frac{4}{x}$

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.

$\frac{x \frac{d V}{d x}}{x} = \frac{V^{2}}{x} + \frac{4}{x}$

Step 6.1.1.2.2.1.2

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

$\frac{d V}{d x} = \frac{V^{2}}{x} + \frac{4}{x}$

Step 6.1.2

Combine the numerators over the common denominator.

$\frac{d V}{d x} = \frac{V^{2} + 4}{x}$

Step 6.1.3

Multiply both sides by $\frac{1}{V^{2} + 4}$ .

$\frac{1}{V^{2} + 4} \frac{d V}{d x} = \frac{1}{V^{2} + 4} \cdot \frac{V^{2} + 4}{x}$

Step 6.1.4

Cancel the common factor of $V^{2} + 4$ .

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Step 6.1.4.1

Cancel the common factor.

$\frac{1}{V^{2} + 4} \frac{d V}{d x} = \frac{1}{V^{2} + 4} \cdot \frac{V^{2} + 4}{x}$

Step 6.1.4.2

Rewrite the expression.

$\frac{1}{V^{2} + 4} \frac{d V}{d x} = \frac{1}{x}$

Step 6.1.5

Rewrite the equation.

$\frac{1}{V^{2} + 4} d V = \frac{1}{x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{1}{V^{2} + 4} d V = \int \frac{1}{x} d x$

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Simplify the expression.

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Step 6.2.2.1.1

Reorder $V^{2}$ and $4$ .

$\int \frac{1}{4 + V^{2}} d V = \int \frac{1}{x} d x$

Step 6.2.2.1.2

Rewrite $4$ as $2^{2}$ .

$\int \frac{1}{2^{2} + V^{2}} d V = \int \frac{1}{x} d x$

Step 6.2.2.2

The integral of $\frac{1}{2^{2} + V^{2}}$ with respect to $V$ is $\frac{1}{2} \arctan (\frac{V}{2}) + C_{1}$ .

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

Step 6.2.2.3

Simplify the answer.

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Step 6.2.2.3.1

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

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

Step 6.2.2.3.2

Rewrite $\frac{\arctan (\frac{V}{2})}{2} + C_{1}$ as $\frac{1}{2} \arctan (\frac{1}{2} V) + C_{1}$ .

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 6.3

Solve for $V$ .

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Step 6.3.1

Multiply each term in $\frac{1}{2} \arctan (\frac{1}{2} V) = \ln (| x |) + C$ by $2$ to eliminate the fractions.

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Step 6.3.1.1

Multiply each term in $\frac{1}{2} \arctan (\frac{1}{2} V) = \ln (| x |) + C$ by $2$ .

$\frac{1}{2} \arctan (\frac{1}{2} V) \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

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

$\frac{1}{2} \arctan (\frac{V}{2}) \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 6.3.1.2.2

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

$\frac{\arctan (\frac{V}{2})}{2} \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 6.3.1.2.3

Cancel the common factor of $2$ .

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Step 6.3.1.2.3.1

Cancel the common factor.

$\frac{\arctan (\frac{V}{2})}{2} \cdot 2 = \ln (| x |) \cdot 2 + C \cdot 2$

Step 6.3.1.2.3.2

Rewrite the expression.

$\arctan (\frac{V}{2}) = \ln (| x |) \cdot 2 + C \cdot 2$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Multiply $\ln (| x |) \cdot 2$ .

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Step 6.3.1.3.1.1.1

Reorder $\ln (| x |)$ and $2$ .

$\arctan (\frac{V}{2}) = 2 \cdot \ln (| x |) + C \cdot 2$

Step 6.3.1.3.1.1.2

Simplify $2 \cdot \ln (| x |)$ by moving $2$ inside the logarithm.

$\arctan (\frac{V}{2}) = \ln ({| x |}^{2}) + C \cdot 2$

Step 6.3.1.3.1.2

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$\arctan (\frac{V}{2}) = \ln (x^{2}) + C \cdot 2$

Step 6.3.1.3.1.3

Move $2$ to the left of $C$ .

$\arctan (\frac{V}{2}) = \ln (x^{2}) + 2 C$

Step 6.3.2

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

$\frac{V}{2} = \tan (\ln (x^{2}) + 2 C)$

Step 6.3.3

Multiply both sides of the equation by $2$ .

$2 \frac{V}{2} = 2 \tan (\ln (x^{2}) + 2 C)$

Step 6.3.4

Simplify the left side.

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Step 6.3.4.1

Cancel the common factor of $2$ .

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Step 6.3.4.1.1

Cancel the common factor.

$2 \frac{V}{2} = 2 \tan (\ln (x^{2}) + 2 C)$

Step 6.3.4.1.2

Rewrite the expression.

$V = 2 \tan (\ln (x^{2}) + 2 C)$

Step 6.4

Simplify the constant of integration.

$V = 2 \tan (\ln (x^{2}) + C)$

Step 7

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

$\frac{y}{x} = 2 \tan (\ln (x^{2}) + C)$

Step 8

Solve $\frac{y}{x} = 2 \tan (\ln (x^{2}) + C)$ for $y$ .

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Step 8.1

Multiply both sides by $x$ .

$\frac{y}{x} x = 2 \tan (\ln (x^{2}) + C) x$

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $x$ .

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Step 8.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = 2 \tan (\ln (x^{2}) + C) x$

Step 8.2.1.1.2

Rewrite the expression.

$y = 2 \tan (\ln (x^{2}) + C) x$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Reorder factors in $2 \tan (\ln (x^{2}) + C) x$ .

$y = 2 x \tan (\ln (x^{2}) + C)$