Calculus Examples

$\frac{d y}{d x} = \frac{- 2 x}{3 y^{2}}$ , $y (2) = 1$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $y^{2}$ .

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

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

Tap for more steps...

Step 1.2.1.1

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

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

Step 1.2.1.2

Cancel the common factor.

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

Step 1.2.1.3

Rewrite the expression.

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

Step 1.2.2

Move the negative in front of the fraction.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

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

$\frac{1}{3} y^{3} + C_{1} = \int - \frac{2 x}{3} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

$\frac{1}{3} y^{3} + C_{1} = - \int \frac{2 x}{3} d x$

Step 2.3.2

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

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

Step 2.3.3

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{2}{3} (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.4

Simplify the answer.

Tap for more steps...

Step 2.3.4.1

Rewrite $- \frac{2}{3} (\frac{1}{2} x^{2} + C_{2})$ as $- \frac{2}{3} \cdot \frac{1}{2} x^{2} + C_{2}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{2}{3} \cdot \frac{1}{2} x^{2} + C_{2}$

Step 2.3.4.2

Simplify.

Tap for more steps...

Step 2.3.4.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{2}{2 \cdot 3} x^{2} + C_{2}$

Step 2.3.4.2.2

Multiply $2$ by $3$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{2}{6} x^{2} + C_{2}$

Step 2.3.4.2.3

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 2.3.4.2.3.1

Factor $2$ out of $2$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{2 (1)}{6} x^{2} + C_{2}$

Step 2.3.4.2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.4.2.3.2.1

Factor $2$ out of $6$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{2 \cdot 1}{2 \cdot 3} x^{2} + C_{2}$

Step 2.3.4.2.3.2.2

Cancel the common factor.

$\frac{1}{3} y^{3} + C_{1} = - \frac{2 \cdot 1}{2 \cdot 3} x^{2} + C_{2}$

Step 2.3.4.2.3.2.3

Rewrite the expression.

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{3} x^{2} + C_{2}$

Step 2.4

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

$\frac{1}{3} y^{3} = - \frac{1}{3} x^{2} + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (- \frac{1}{3} x^{2} + K)$

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

Simplify $3 (\frac{1}{3} y^{3})$ .

Tap for more steps...

Step 3.2.1.1.1

Combine $\frac{1}{3}$ and $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (- \frac{1}{3} x^{2} + K)$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

$3 \frac{y^{3}}{3} = 3 (- \frac{1}{3} x^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (- \frac{1}{3} x^{2} + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $3 (- \frac{1}{3} x^{2} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{3}$ into the numerator.

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

Step 3.2.2.1.3.2

Cancel the common factor.

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

Step 3.2.2.1.3.3

Rewrite the expression.

$y^{3} = - x^{2} + 3 K$

Step 3.3

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

$y = \sqrt[3]{- x^{2} + 3 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{- x^{2} + K}$

Step 5

Use the initial condition to find the value of $K$ by substituting $2$ for $x$ and $1$ for $y$ in $y = \sqrt[3]{- x^{2} + K}$ .

$1 = \sqrt[3]{- 2^{2} + K}$

Step 6

Solve for $K$ .

Tap for more steps...

Step 6.1

Rewrite the equation as $\sqrt[3]{- 2^{2} + K} = 1$ .

$\sqrt[3]{- 2^{2} + K} = 1$

Step 6.2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{- 2^{2} + K}}^{3} = 1^{3}$

Step 6.3

Simplify each side of the equation.

Tap for more steps...

Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- 2^{2} + K}$ as ${(- 2^{2} + K)}^{\frac{1}{3}}$ .

${({(- 2^{2} + K)}^{\frac{1}{3}})}^{3} = 1^{3}$

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Simplify ${({(- 2^{2} + K)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 6.3.2.1.1

Multiply the exponents in ${({(- 2^{2} + K)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 6.3.2.1.1.1

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

${(- 2^{2} + K)}^{\frac{1}{3} \cdot 3} = 1^{3}$

Step 6.3.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.2.1.1.2.1

Cancel the common factor.

${(- 2^{2} + K)}^{\frac{1}{3} \cdot 3} = 1^{3}$

Step 6.3.2.1.1.2.2

Rewrite the expression.

${(- 2^{2} + K)}^{1} = 1^{3}$

Step 6.3.2.1.2

Simplify each term.

Tap for more steps...

Step 6.3.2.1.2.1

Raise $2$ to the power of $2$ .

${(- 1 \cdot 4 + K)}^{1} = 1^{3}$

Step 6.3.2.1.2.2

Multiply $- 1$ by $4$ .

${(- 4 + K)}^{1} = 1^{3}$

Step 6.3.2.1.3

Simplify.

$- 4 + K = 1^{3}$

Step 6.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.1

One to any power is one.

$- 4 + K = 1$

Step 6.4

Move all terms not containing $K$ to the right side of the equation.

Tap for more steps...

Step 6.4.1

Add $4$ to both sides of the equation.

$K = 1 + 4$

Step 6.4.2

Add $1$ and $4$ .

$K = 5$

Step 7

Substitute $5$ for $K$ in $y = \sqrt[3]{- x^{2} + K}$ and simplify.

Tap for more steps...

Step 7.1

Substitute $5$ for $K$ .

$y = \sqrt[3]{- x^{2} + 5}$