Calculus Examples

$\frac{d y}{d x} = \frac{x^{3}}{y^{2}}$ , $y (2) = \sqrt[3]{15}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$y^{2} d y = x^{3} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{4} x^{4} + C_{2}$

Step 2.4

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

$\frac{1}{3} y^{3} = \frac{1}{4} x^{4} + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\frac{1}{4} x^{4} + K)$

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 $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (\frac{1}{4} x^{4} + K)$

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.

$3 \frac{y^{3}}{3} = 3 (\frac{1}{4} x^{4} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\frac{1}{4} x^{4} + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $3 (\frac{1}{4} x^{4} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$y^{3} = 3 (\frac{x^{4}}{4} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = 3 \frac{x^{4}}{4} + 3 K$

Step 3.2.2.1.3

Combine $3$ and $\frac{x^{4}}{4}$ .

$y^{3} = \frac{3 x^{4}}{4} + 3 K$

Step 3.3

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

$y = \sqrt[3]{\frac{3 x^{4}}{4} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{\frac{3 x^{4}}{4} + 3 K}$ .

Tap for more steps...

Step 3.4.1

Factor $3$ out of $\frac{3 x^{4}}{4} + 3 K$ .

Tap for more steps...

Step 3.4.1.1

Factor $3$ out of $\frac{3 x^{4}}{4}$ .

$y = \sqrt[3]{3 \frac{x^{4}}{4} + 3 K}$

Step 3.4.1.2

Factor $3$ out of $3 K$ .

$y = \sqrt[3]{3 \frac{x^{4}}{4} + 3 K}$

Step 3.4.1.3

Factor $3$ out of $3 \frac{x^{4}}{4} + 3 K$ .

$y = \sqrt[3]{3 (\frac{x^{4}}{4} + K)}$

Step 3.4.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \sqrt[3]{3 (\frac{x^{4}}{4} + K \cdot \frac{4}{4})}$

Step 3.4.3

Simplify terms.

Tap for more steps...

Step 3.4.3.1

Combine $K$ and $\frac{4}{4}$ .

$y = \sqrt[3]{3 (\frac{x^{4}}{4} + \frac{K \cdot 4}{4})}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{3 \frac{x^{4} + K \cdot 4}{4}}$

Step 3.4.4

Move $4$ to the left of $K$ .

$y = \sqrt[3]{3 \frac{x^{4} + 4 K}{4}}$

Step 3.4.5

Combine $3$ and $\frac{x^{4} + 4 K}{4}$ .

$y = \sqrt[3]{\frac{3 (x^{4} + 4 K)}{4}}$

Step 3.4.6

Rewrite $\sqrt[3]{\frac{3 (x^{4} + 4 K)}{4}}$ as $\frac{\sqrt[3]{3 (x^{4} + 4 K)}}{\sqrt[3]{4}}$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)}}{\sqrt[3]{4}}$

Step 3.4.7

Multiply $\frac{\sqrt[3]{3 (x^{4} + 4 K)}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 3.4.8

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.8.1

Multiply $\frac{\sqrt[3]{3 (x^{4} + 4 K)}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 3.4.8.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.4.8.3

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

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.4.8.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 3.4.8.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

Tap for more steps...

Step 3.4.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.4.8.5.2

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

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.4.8.5.3

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

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.8.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.4.8.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.8.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.4.8.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4}$

Step 3.4.9

Simplify the numerator.

Tap for more steps...

Step 3.4.9.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} \sqrt[3]{4^{2}}}{4}$

Step 3.4.9.2

Raise $4$ to the power of $2$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} \sqrt[3]{16}}{4}$

Step 3.4.9.3

Rewrite $16$ as $2^{3} \cdot 2$ .

Tap for more steps...

Step 3.4.9.3.1

Factor $8$ out of $16$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} \sqrt[3]{8 (2)}}{4}$

Step 3.4.9.3.2

Rewrite $8$ as $2^{3}$ .

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 3.4.9.4

Pull terms out from under the radical.

$y = \frac{\sqrt[3]{3 (x^{4} + 4 K)} \cdot 2 \sqrt[3]{2}}{4}$

Step 3.4.9.5

Combine exponents.

Tap for more steps...

Step 3.4.9.5.1

Combine using the product rule for radicals.

$y = \frac{2 \sqrt[3]{3 (x^{4} + 4 K) \cdot 2}}{4}$

Step 3.4.9.5.2

Multiply $2$ by $3$ .

$y = \frac{2 \sqrt[3]{6 (x^{4} + 4 K)}}{4}$

Step 3.4.10

Cancel the common factors.

Tap for more steps...

Step 3.4.10.1

Factor $2$ out of $4$ .

$y = \frac{2 \sqrt[3]{6 (x^{4} + 4 K)}}{2 \cdot 2}$

Step 3.4.10.2

Cancel the common factor.

$y = \frac{2 \sqrt[3]{6 (x^{4} + 4 K)}}{2 \cdot 2}$

Step 3.4.10.3

Rewrite the expression.

$y = \frac{\sqrt[3]{6 (x^{4} + 4 K)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{6 (x^{4} + K)}}{2}$

Step 5

Use the initial condition to find the value of $K$ by substituting $2$ for $x$ and $\sqrt[3]{15}$ for $y$ in $y = \frac{\sqrt[3]{6 (x^{4} + K)}}{2}$ .

$\sqrt[3]{15} = \frac{\sqrt[3]{6 (2^{4} + K)}}{2}$

Step 6

Solve for $K$ .

Tap for more steps...

Step 6.1

Rewrite the equation as $\frac{\sqrt[3]{6 (2^{4} + K)}}{2} = \sqrt[3]{15}$ .

$\frac{\sqrt[3]{6 (2^{4} + K)}}{2} = \sqrt[3]{15}$

Step 6.2

Multiply both sides by $2$ .

$\frac{\sqrt[3]{6 (2^{4} + K)}}{2} \cdot 2 = \sqrt[3]{15} \cdot 2$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Simplify the left side.

Tap for more steps...

Step 6.3.1.1

Simplify $\frac{\sqrt[3]{6 (2^{4} + K)}}{2} \cdot 2$ .

Tap for more steps...

Step 6.3.1.1.1

Raise $2$ to the power of $4$ .

$\frac{\sqrt[3]{6 (16 + K)}}{2} \cdot 2 = \sqrt[3]{15} \cdot 2$

Step 6.3.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.1.1.2.1

Cancel the common factor.

$\frac{\sqrt[3]{6 (16 + K)}}{2} \cdot 2 = \sqrt[3]{15} \cdot 2$

Step 6.3.1.1.2.2

Rewrite the expression.

$\sqrt[3]{6 (16 + K)} = \sqrt[3]{15} \cdot 2$

Step 6.3.2

Simplify the right side.

Tap for more steps...

Step 6.3.2.1

Move $2$ to the left of $\sqrt[3]{15}$ .

$\sqrt[3]{6 (16 + K)} = 2 \sqrt[3]{15}$

Step 6.4

Solve for $K$ .

Tap for more steps...

Step 6.4.1

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

${\sqrt[3]{6 (16 + K)}}^{3} = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2

Simplify each side of the equation.

Tap for more steps...

Step 6.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 (16 + K)}$ as ${(6 (16 + K))}^{\frac{1}{3}}$ .

${({(6 (16 + K))}^{\frac{1}{3}})}^{3} = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.2.1

Simplify ${({(6 (16 + K))}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 6.4.2.2.1.1

Multiply the exponents in ${({(6 (16 + K))}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 6.4.2.2.1.1.1

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

${(6 (16 + K))}^{\frac{1}{3} \cdot 3} = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.4.2.2.1.1.2.1

Cancel the common factor.

${(6 (16 + K))}^{\frac{1}{3} \cdot 3} = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2.2.1.1.2.2

Rewrite the expression.

${(6 (16 + K))}^{1} = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2.2.1.2

Apply the distributive property.

${(6 \cdot 16 + 6 K)}^{1} = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2.2.1.3

Multiply.

Tap for more steps...

Step 6.4.2.2.1.3.1

Multiply $6$ by $16$ .

${(96 + 6 K)}^{1} = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2.2.1.3.2

Simplify.

$96 + 6 K = {(2 \sqrt[3]{15})}^{3}$

Step 6.4.2.3

Simplify the right side.

Tap for more steps...

Step 6.4.2.3.1

Simplify ${(2 \sqrt[3]{15})}^{3}$ .

Tap for more steps...

Step 6.4.2.3.1.1

Simplify the expression.

Tap for more steps...

Step 6.4.2.3.1.1.1

Apply the product rule to $2 \sqrt[3]{15}$ .

$96 + 6 K = 2^{3} {\sqrt[3]{15}}^{3}$

Step 6.4.2.3.1.1.2

Raise $2$ to the power of $3$ .

$96 + 6 K = 8 {\sqrt[3]{15}}^{3}$

Step 6.4.2.3.1.2

Rewrite ${\sqrt[3]{15}}^{3}$ as $15$ .

Tap for more steps...

Step 6.4.2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{15}$ as $15^{\frac{1}{3}}$ .

$96 + 6 K = 8 {(15^{\frac{1}{3}})}^{3}$

Step 6.4.2.3.1.2.2

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

$96 + 6 K = 8 \cdot 15^{\frac{1}{3} \cdot 3}$

Step 6.4.2.3.1.2.3

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

$96 + 6 K = 8 \cdot 15^{\frac{3}{3}}$

Step 6.4.2.3.1.2.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.4.2.3.1.2.4.1

Cancel the common factor.

$96 + 6 K = 8 \cdot 15^{\frac{3}{3}}$

Step 6.4.2.3.1.2.4.2

Rewrite the expression.

$96 + 6 K = 8 \cdot 15^{1}$

Step 6.4.2.3.1.2.5

Evaluate the exponent.

$96 + 6 K = 8 \cdot 15$

Step 6.4.2.3.1.3

Multiply $8$ by $15$ .

$96 + 6 K = 120$

Step 6.4.3

Solve for $K$ .

Tap for more steps...

Step 6.4.3.1

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

Tap for more steps...

Step 6.4.3.1.1

Subtract $96$ from both sides of the equation.

$6 K = 120 - 96$

Step 6.4.3.1.2

Subtract $96$ from $120$ .

$6 K = 24$

Step 6.4.3.2

Divide each term in $6 K = 24$ by $6$ and simplify.

Tap for more steps...

Step 6.4.3.2.1

Divide each term in $6 K = 24$ by $6$ .

$\frac{6 K}{6} = \frac{24}{6}$

Step 6.4.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.4.3.2.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 6.4.3.2.2.1.1

Cancel the common factor.

$\frac{6 K}{6} = \frac{24}{6}$

Step 6.4.3.2.2.1.2

Divide $K$ by $1$ .

$K = \frac{24}{6}$

Step 6.4.3.2.3

Simplify the right side.

Tap for more steps...

Step 6.4.3.2.3.1

Divide $24$ by $6$ .

$K = 4$

Step 7

Substitute $4$ for $K$ in $y = \frac{\sqrt[3]{6 (x^{4} + K)}}{2}$ and simplify.

Tap for more steps...

Step 7.1

Substitute $4$ for $K$ .

$y = \frac{\sqrt[3]{6 (x^{4} + 4)}}{2}$