Calculus Examples

$x \frac{d y}{d x} = y + 2 \sqrt{x y}$

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

$\frac{x \frac{d y}{d x}}{x} = \frac{y}{x} + \frac{2 \sqrt{x y}}{x}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} = \frac{y}{x} + \frac{2 \sqrt{x y}}{x}$

Step 1.1.2.1.2

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

$\frac{d y}{d x} = \frac{y}{x} + \frac{2 \sqrt{x y}}{x}$

Step 1.2

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

$\frac{d y}{d x} = \frac{y}{x} + 2 \frac{\sqrt{x y}}{x}$

Step 1.3

Assume $\sqrt{x^{2}} = x$ .

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

Step 1.4

Combine $\sqrt{x y}$ and $\sqrt{x^{2}}$ into a single radical.

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

Step 1.5

Reduce the expression $\frac{x y}{x^{2}}$ by cancelling the common factors.

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Step 1.5.1

Factor $x$ out of $x y$ .

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

Step 1.5.2

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

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

Step 1.5.3

Cancel the common factor.

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

Step 1.5.4

Rewrite the expression.

$\frac{d y}{d x} = \frac{y}{x} + 2 \sqrt{\frac{y}{x}}$

Step 2

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

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

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

$x \frac{d V}{d x} + V = V + 2 \sqrt{V}$

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.

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

Step 6.1.1.1.2

Combine the opposite terms in $V + 2 \sqrt{V} - V$ .

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Step 6.1.1.1.2.1

Subtract $V$ from $V$ .

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

Step 6.1.1.1.2.2

Add $0$ and $2 \sqrt{V}$ .

$x \frac{d V}{d x} = 2 \sqrt{V}$

Step 6.1.1.2

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

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Step 6.1.1.2.1

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

$\frac{x \frac{d V}{d x}}{x} = \frac{2 \sqrt{V}}{x}$

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.

$\frac{x \frac{d V}{d x}}{x} = \frac{2 \sqrt{V}}{x}$

Step 6.1.1.2.2.1.2

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

$\frac{d V}{d x} = \frac{2 \sqrt{V}}{x}$

Step 6.1.2

Multiply both sides by $\frac{1}{\sqrt{V}}$ .

$\frac{1}{\sqrt{V}} \frac{d V}{d x} = \frac{1}{\sqrt{V}} \cdot \frac{2 \sqrt{V}}{x}$

Step 6.1.3

Cancel the common factor of $\sqrt{V}$ .

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Step 6.1.3.1

Factor $\sqrt{V}$ out of $2 \sqrt{V}$ .

$\frac{1}{\sqrt{V}} \frac{d V}{d x} = \frac{1}{\sqrt{V}} \cdot \frac{\sqrt{V} \cdot 2}{x}$

Step 6.1.3.2

Cancel the common factor.

$\frac{1}{\sqrt{V}} \frac{d V}{d x} = \frac{1}{\sqrt{V}} \cdot \frac{\sqrt{V} \cdot 2}{x}$

Step 6.1.3.3

Rewrite the expression.

$\frac{1}{\sqrt{V}} \frac{d V}{d x} = \frac{2}{x}$

Step 6.1.4

Rewrite the equation.

$\frac{1}{\sqrt{V}} d V = \frac{2}{x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

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

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Apply basic rules of exponents.

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Step 6.2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{V}$ as $V^{\frac{1}{2}}$ .

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

Step 6.2.2.1.2

Move $V^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 6.2.2.1.3

Multiply the exponents in ${(V^{\frac{1}{2}})}^{- 1}$ .

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Step 6.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.2.2.1.3.2

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

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

Step 6.2.2.1.3.3

Move the negative in front of the fraction.

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

Step 6.2.2.2

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

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

Step 6.2.3

Integrate the right side.

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

Simplify.

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

Step 6.2.4

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

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

Step 6.3

Solve for $V$ .

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Step 6.3.1

Divide each term in $2 V^{\frac{1}{2}} = 2 \ln (| x |) + C$ by $2$ and simplify.

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Step 6.3.1.1

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

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

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor.

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

Step 6.3.1.2.2

Divide $V^{\frac{1}{2}}$ by $1$ .

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

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Cancel the common factor of $2$ .

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Step 6.3.1.3.1.1

Cancel the common factor.

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

Step 6.3.1.3.1.2

Divide $\ln (| x |)$ by $1$ .

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

Step 6.3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 6.3.3

Simplify the left side.

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Step 6.3.3.1

Simplify ${(V^{\frac{1}{2}})}^{2}$ .

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Step 6.3.3.1.1

Multiply the exponents in ${(V^{\frac{1}{2}})}^{2}$ .

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Step 6.3.3.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.3.3.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.3.1.1.2.1

Cancel the common factor.

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

Step 6.3.3.1.1.2.2

Rewrite the expression.

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

Step 6.3.3.1.2

Simplify.

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

Step 6.4

Simplify the constant of integration.

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

Step 7

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

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

Step 8

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

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Step 8.1

Multiply both sides by $x$ .

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

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.

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

Step 8.2.1.1.2

Rewrite the expression.

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

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

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

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