Calculus Examples

$\frac{d y}{d x} = \frac{2 x^{2} y + y^{3}}{2 x^{3}}$

Step 1

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

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Step 1.1

Split $\frac{2 x^{2} y + y^{3}}{2 x^{3}}$ and simplify.

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Step 1.1.1

Split the fraction $\frac{2 x^{2} y + y^{3}}{2 x^{3}}$ into two fractions.

$\frac{d y}{d x} = \frac{2 x^{2} y}{2 x^{3}} + \frac{y^{3}}{2 x^{3}}$

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{2 x^{2} y}{2 x^{3}} + \frac{y^{3}}{2 x^{3}}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{d y}{d x} = \frac{x^{2} y}{x^{3}} + \frac{y^{3}}{2 x^{3}}$

Step 1.1.2.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 1.1.2.2.1

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

$\frac{d y}{d x} = \frac{x^{2} (y)}{x^{3}} + \frac{y^{3}}{2 x^{3}}$

Step 1.1.2.2.2

Cancel the common factors.

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Step 1.1.2.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{d y}{d x} = \frac{x^{2} (y)}{x^{2} x} + \frac{y^{3}}{2 x^{3}}$

Step 1.1.2.2.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{x^{2} y}{x^{2} x} + \frac{y^{3}}{2 x^{3}}$

Step 1.1.2.2.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{y}{x} + \frac{y^{3}}{2 x^{3}}$

Step 1.2

Factor out $\frac{y}{x}$ from $\frac{y^{3}}{2 x^{3}}$ .

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Step 1.2.1

Factor $\frac{y}{x}$ out of $\frac{y^{3}}{2 x^{3}}$ .

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

Step 1.2.2

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

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

Step 1.3

Factor out $\frac{y}{x}$ from $\frac{y^{2}}{2 x^{2}}$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.4

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

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Step 1.4.1

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

$\frac{d y}{d x} = \frac{y}{x} + \frac{y}{x} \cdot \frac{1}{2} \frac{y}{x} \frac{y}{x}$

Step 1.4.2

Reorder $\frac{y}{x}$ and $\frac{1}{2}$ .

$\frac{d y}{d x} = \frac{y}{x} + \frac{1}{2} \cdot \frac{y}{x} \frac{y}{x} \frac{y}{x}$

Step 2

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

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

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

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

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

Simplify each term.

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Step 6.1.1.1.1

Multiply $V$ by $V$ by adding the exponents.

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Step 6.1.1.1.1.1

Move $V$ .

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

Step 6.1.1.1.1.2

Multiply $V$ by $V$ .

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

Step 6.1.1.1.2

Multiply $V^{2}$ by $V$ by adding the exponents.

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Step 6.1.1.1.2.1

Move $V$ .

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

Step 6.1.1.1.2.2

Multiply $V$ by $V^{2}$ .

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Step 6.1.1.1.2.2.1

Raise $V$ to the power of $1$ .

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

Step 6.1.1.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.1.1.1.2.3

Add $1$ and $2$ .

$x \frac{d V}{d x} + V = V + \frac{1}{2} V^{3}$

Step 6.1.1.1.3

Combine $\frac{1}{2}$ and $V^{3}$ .

$x \frac{d V}{d x} + V = V + \frac{V^{3}}{2}$

Step 6.1.1.2

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

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Step 6.1.1.2.1

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = V + \frac{V^{3}}{2} - V$

Step 6.1.1.2.2

Combine the opposite terms in $V + \frac{V^{3}}{2} - V$ .

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Step 6.1.1.2.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = \frac{V^{3}}{2} + 0$

Step 6.1.1.2.2.2

Add $\frac{V^{3}}{2}$ and $0$ .

$x \frac{d V}{d x} = \frac{V^{3}}{2}$

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = \frac{V^{3}}{2}$ by $x$ and simplify.

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Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \frac{V^{3}}{2}$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{V^{3}}{2}}{x}$

Step 6.1.1.3.2

Simplify the left side.

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Step 6.1.1.3.2.1

Cancel the common factor of $x$ .

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Step 6.1.1.3.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{V^{3}}{2}}{x}$

Step 6.1.1.3.2.1.2

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

$\frac{d V}{d x} = \frac{\frac{V^{3}}{2}}{x}$

Step 6.1.1.3.3

Simplify the right side.

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Step 6.1.1.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d V}{d x} = \frac{V^{3}}{2} \cdot \frac{1}{x}$

Step 6.1.1.3.3.2

Combine.

$\frac{d V}{d x} = \frac{V^{3} \cdot 1}{2 x}$

Step 6.1.1.3.3.3

Multiply $V^{3}$ by $1$ .

$\frac{d V}{d x} = \frac{V^{3}}{2 x}$

Step 6.1.2

Multiply both sides by $\frac{1}{V^{3}}$ .

$\frac{1}{V^{3}} \frac{d V}{d x} = \frac{1}{V^{3}} \cdot \frac{V^{3}}{2 x}$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Combine.

$\frac{1}{V^{3}} \frac{d V}{d x} = \frac{1 V^{3}}{V^{3} (2 x)}$

Step 6.1.3.2

Cancel the common factor of $V^{3}$ .

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Step 6.1.3.2.1

Cancel the common factor.

$\frac{1}{V^{3}} \frac{d V}{d x} = \frac{1 V^{3}}{V^{3} (2 x)}$

Step 6.1.3.2.2

Rewrite the expression.

$\frac{1}{V^{3}} \frac{d V}{d x} = \frac{1}{2 x}$

Step 6.1.4

Rewrite the equation.

$\frac{1}{V^{3}} d V = \frac{1}{2 x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{1}{V^{3}} d V = \int \frac{1}{2 x} d x$

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

Move $V^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 6.2.2.1.2

Multiply the exponents in ${(V^{3})}^{- 1}$ .

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Step 6.2.2.1.2.1

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

$\int V^{3 \cdot - 1} d V = \int \frac{1}{2 x} d x$

Step 6.2.2.1.2.2

Multiply $3$ by $- 1$ .

$\int V^{- 3} d V = \int \frac{1}{2 x} d x$

Step 6.2.2.2

By the Power Rule, the integral of $V^{- 3}$ with respect to $V$ is $- \frac{1}{2} V^{- 2}$ .

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

Step 6.2.2.3

Simplify the answer.

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Step 6.2.2.3.1

Rewrite $- \frac{1}{2} V^{- 2} + C_{1}$ as $- \frac{1}{2} \cdot \frac{1}{V^{2}} + C_{1}$ .

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

Step 6.2.2.3.2

Simplify.

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Step 6.2.2.3.2.1

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

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

Step 6.2.2.3.2.2

Move $2$ to the left of $V^{2}$ .

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

Step 6.2.3

Integrate the right side.

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

Simplify.

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

Step 6.2.4

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

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

Step 6.3

Solve for $V$ .

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Step 6.3.1

Simplify $\frac{1}{2} \ln (| x |)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 6.3.2

Find the LCD of the terms in the equation.

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Step 6.3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 V^{2}, 1, 1$

Step 6.3.2.2

The LCM of one and any expression is the expression.

$2 V^{2}$

Step 6.3.3

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

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Step 6.3.3.1

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

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

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

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

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Step 6.3.3.2.1.1

Move the leading negative in $- \frac{1}{2 V^{2}}$ into the numerator.

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

Step 6.3.3.2.1.2

Cancel the common factor.

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

Step 6.3.3.2.1.3

Rewrite the expression.

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

Step 6.3.3.3

Simplify the right side.

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Step 6.3.3.3.1

Simplify each term.

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Step 6.3.3.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.3.3.1.2

Simplify $2 \ln ({| x |}^{\frac{1}{2}})$ by moving $2$ inside the logarithm.

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

Step 6.3.3.3.1.3

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

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Step 6.3.3.3.1.3.1

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

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

Step 6.3.3.3.1.3.2

Cancel the common factor of $2$ .

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Step 6.3.3.3.1.3.2.1

Cancel the common factor.

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

Step 6.3.3.3.1.3.2.2

Rewrite the expression.

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

Step 6.3.3.3.1.4

Simplify.

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

Step 6.3.3.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 6.3.3.3.2

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

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

Step 6.3.4

Solve the equation.

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Step 6.3.4.1

Rewrite the equation as $V^{2} \ln (| x |) + 2 C V^{2} = - 1$ .

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

Step 6.3.4.2

Factor $V^{2}$ out of $V^{2} \ln (| x |) + 2 C V^{2}$ .

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Step 6.3.4.2.1

Factor $V^{2}$ out of $2 C V^{2}$ .

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

Step 6.3.4.2.2

Factor $V^{2}$ out of $V^{2} \ln (| x |) + V^{2} (2 C)$ .

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

Step 6.3.4.3

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

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Step 6.3.4.3.1

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

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

Step 6.3.4.3.2

Simplify the left side.

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Step 6.3.4.3.2.1

Cancel the common factor of $\ln (| x |) + 2 C$ .

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Step 6.3.4.3.2.1.1

Cancel the common factor.

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

Step 6.3.4.3.2.1.2

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

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

Step 6.3.4.3.3

Simplify the right side.

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Step 6.3.4.3.3.1

Move the negative in front of the fraction.

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

Step 6.3.4.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 6.3.4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.4.5.1

First, use the positive value of the $\pm$ to find the first solution.