Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $y$ .

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.2.2

Rewrite ${(3 - x)}^{2}$ as $(3 - x) (3 - x)$ .

$y \frac{d y}{d x} = (3 - x) (3 - x)$

Step 1.2.3

Expand $(3 - x) (3 - x)$ using the FOIL Method.

Tap for more steps...

Step 1.2.3.1

Apply the distributive property.

$y \frac{d y}{d x} = 3 (3 - x) - x (3 - x)$

Step 1.2.3.2

Apply the distributive property.

$y \frac{d y}{d x} = 3 \cdot 3 + 3 (- x) - x (3 - x)$

Step 1.2.3.3

Apply the distributive property.

$y \frac{d y}{d x} = 3 \cdot 3 + 3 (- x) - x \cdot 3 - x (- x)$

Step 1.2.4

Simplify and combine like terms.

Tap for more steps...

Step 1.2.4.1

Simplify each term.

Tap for more steps...

Step 1.2.4.1.1

Multiply $3$ by $3$ .

$y \frac{d y}{d x} = 9 + 3 (- x) - x \cdot 3 - x (- x)$

Step 1.2.4.1.2

Multiply $- 1$ by $3$ .

$y \frac{d y}{d x} = 9 - 3 x - x \cdot 3 - x (- x)$

Step 1.2.4.1.3

Multiply $3$ by $- 1$ .

$y \frac{d y}{d x} = 9 - 3 x - 3 x - x (- x)$

Step 1.2.4.1.4

Rewrite using the commutative property of multiplication.

$y \frac{d y}{d x} = 9 - 3 x - 3 x - 1 \cdot - 1 x \cdot x$

Step 1.2.4.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.2.4.1.5.1

Move $x$ .

$y \frac{d y}{d x} = 9 - 3 x - 3 x - 1 \cdot - 1 (x \cdot x)$

Step 1.2.4.1.5.2

Multiply $x$ by $x$ .

$y \frac{d y}{d x} = 9 - 3 x - 3 x - 1 \cdot - 1 x^{2}$

Step 1.2.4.1.6

Multiply $- 1$ by $- 1$ .

$y \frac{d y}{d x} = 9 - 3 x - 3 x + 1 x^{2}$

Step 1.2.4.1.7

Multiply $x^{2}$ by $1$ .

$y \frac{d y}{d x} = 9 - 3 x - 3 x + x^{2}$

Step 1.2.4.2

Subtract $3 x$ from $- 3 x$ .

$y \frac{d y}{d x} = 9 - 6 x + x^{2}$

Step 1.3

Rewrite the equation.

$y d y = (9 - 6 x + x^{2}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int y d y = \int 9 - 6 x + x^{2} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int 9 - 6 x + x^{2} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$\frac{1}{2} y^{2} + C_{1} = \int 9 d x + \int - 6 x d x + \int x^{2} d x$

Step 2.3.2

Apply the constant rule.

$\frac{1}{2} y^{2} + C_{1} = 9 x + C_{2} + \int - 6 x d x + \int x^{2} d x$

Step 2.3.3

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = 9 x + C_{2} - 6 \int x d x + \int x^{2} d x$

Step 2.3.4

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

$\frac{1}{2} y^{2} + C_{1} = 9 x + C_{2} - 6 (\frac{1}{2} x^{2} + C_{3}) + \int x^{2} d x$

Step 2.3.5

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

$\frac{1}{2} y^{2} + C_{1} = 9 x + C_{2} - 6 (\frac{1}{2} x^{2} + C_{3}) + \frac{1}{3} x^{3} + C_{4}$

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

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

$\frac{1}{2} y^{2} + C_{1} = 9 x + C_{2} - 6 (\frac{x^{2}}{2} + C_{3}) + \frac{1}{3} x^{3} + C_{4}$

Step 2.3.6.2

Simplify.

$\frac{1}{2} y^{2} + C_{1} = 9 x - 3 x^{2} + \frac{1}{3} x^{3} + C_{5}$

Step 2.3.7

Reorder terms.

$\frac{1}{2} y^{2} + C_{1} = - 3 x^{2} + \frac{1}{3} x^{3} + 9 x + C_{5}$

Step 2.4

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

$\frac{1}{2} y^{2} = - 3 x^{2} + \frac{1}{3} x^{3} + 9 x + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- 3 x^{2} + \frac{1}{3} x^{3} + 9 x + K)$

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 $2 (\frac{1}{2} y^{2})$ .

Tap for more steps...

Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (- 3 x^{2} + \frac{1}{3} x^{3} + 9 x + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (- 3 x^{2} + \frac{1}{3} x^{3} + 9 x + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- 3 x^{2} + \frac{1}{3} x^{3} + 9 x + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (- 3 x^{2} + \frac{1}{3} x^{3} + 9 x + K)$ .

Tap for more steps...

Step 3.2.2.1.1

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

$y^{2} = 2 (- 3 x^{2} + \frac{x^{3}}{3} + 9 x + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 (- 3 x^{2}) + 2 \frac{x^{3}}{3} + 2 (9 x) + 2 K$

Step 3.2.2.1.3

Simplify.

Tap for more steps...

Step 3.2.2.1.3.1

Multiply $- 3$ by $2$ .

$y^{2} = - 6 x^{2} + 2 \frac{x^{3}}{3} + 2 (9 x) + 2 K$

Step 3.2.2.1.3.2

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

$y^{2} = - 6 x^{2} + \frac{2 x^{3}}{3} + 2 (9 x) + 2 K$

Step 3.2.2.1.3.3

Multiply $9$ by $2$ .

$y^{2} = - 6 x^{2} + \frac{2 x^{3}}{3} + 18 x + 2 K$

Step 3.3

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

$y = \pm \sqrt{- 6 x^{2} + \frac{2 x^{3}}{3} + 18 x + 2 K}$

Step 3.4

Simplify $\pm \sqrt{- 6 x^{2} + \frac{2 x^{3}}{3} + 18 x + 2 K}$ .

Tap for more steps...

Step 3.4.1

Factor $2$ out of $- 6 x^{2} + \frac{2 x^{3}}{3} + 18 x + 2 K$ .

Tap for more steps...

Step 3.4.1.1

Factor $2$ out of $- 6 x^{2}$ .

$y = \pm \sqrt{2 (- 3 x^{2}) + \frac{2 x^{3}}{3} + 18 x + 2 K}$

Step 3.4.1.2

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

$y = \pm \sqrt{2 (- 3 x^{2}) + 2 \frac{x^{3}}{3} + 18 x + 2 K}$

Step 3.4.1.3

Factor $2$ out of $18 x$ .

$y = \pm \sqrt{2 (- 3 x^{2}) + 2 \frac{x^{3}}{3} + 2 (9 x) + 2 K}$

Step 3.4.1.4

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (- 3 x^{2}) + 2 \frac{x^{3}}{3} + 2 (9 x) + 2 K}$

Step 3.4.1.5

Factor $2$ out of $2 (- 3 x^{2}) + 2 \frac{x^{3}}{3}$ .

$y = \pm \sqrt{2 (- 3 x^{2} + \frac{x^{3}}{3}) + 2 (9 x) + 2 K}$

Step 3.4.1.6

Factor $2$ out of $2 (- 3 x^{2} + \frac{x^{3}}{3}) + 2 (9 x)$ .

$y = \pm \sqrt{2 (- 3 x^{2} + \frac{x^{3}}{3} + 9 x) + 2 K}$

Step 3.4.1.7

Factor $2$ out of $2 (- 3 x^{2} + \frac{x^{3}}{3} + 9 x) + 2 K$ .

$y = \pm \sqrt{2 (- 3 x^{2} + \frac{x^{3}}{3} + 9 x + K)}$

Step 3.4.2

To write $- 3 x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{2 (- 3 x^{2} \cdot \frac{3}{3} + \frac{x^{3}}{3} + 9 x + K)}$

Step 3.4.3

Simplify terms.

Tap for more steps...

Step 3.4.3.1

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

$y = \pm \sqrt{2 (\frac{- 3 x^{2} \cdot 3}{3} + \frac{x^{3}}{3} + 9 x + K)}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{2 (\frac{- 3 x^{2} \cdot 3 + x^{3}}{3} + 9 x + K)}$

Step 3.4.4

Simplify the numerator.

Tap for more steps...

Step 3.4.4.1

Factor $x^{2}$ out of $- 3 x^{2} \cdot 3 + x^{3}$ .

Tap for more steps...

Step 3.4.4.1.1

Factor $x^{2}$ out of $- 3 x^{2} \cdot 3$ .

$y = \pm \sqrt{2 (\frac{x^{2} (- 3 \cdot 3) + x^{3}}{3} + 9 x + K)}$

Step 3.4.4.1.2

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

$y = \pm \sqrt{2 (\frac{x^{2} (- 3 \cdot 3) + x^{2} x}{3} + 9 x + K)}$

Step 3.4.4.1.3

Factor $x^{2}$ out of $x^{2} (- 3 \cdot 3) + x^{2} x$ .

$y = \pm \sqrt{2 (\frac{x^{2} (- 3 \cdot 3 + x)}{3} + 9 x + K)}$

Step 3.4.4.2

Multiply $- 3$ by $3$ .

$y = \pm \sqrt{2 (\frac{x^{2} (- 9 + x)}{3} + 9 x + K)}$

Step 3.4.5

To write $9 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{2 (\frac{x^{2} (- 9 + x)}{3} + 9 x \cdot \frac{3}{3} + K)}$

Step 3.4.6

Simplify terms.

Tap for more steps...

Step 3.4.6.1

Combine $9 x$ and $\frac{3}{3}$ .

$y = \pm \sqrt{2 (\frac{x^{2} (- 9 + x)}{3} + \frac{9 x \cdot 3}{3} + K)}$

Step 3.4.6.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{2 (\frac{x^{2} (- 9 + x) + 9 x \cdot 3}{3} + K)}$

Step 3.4.7

Simplify the numerator.

Tap for more steps...

Step 3.4.7.1

Factor $x$ out of $x^{2} (- 9 + x) + 9 x \cdot 3$ .

Tap for more steps...

Step 3.4.7.1.1

Factor $x$ out of $x^{2} (- 9 + x)$ .

$y = \pm \sqrt{2 (\frac{x (x (- 9 + x)) + 9 x \cdot 3}{3} + K)}$

Step 3.4.7.1.2

Factor $x$ out of $9 x \cdot 3$ .

$y = \pm \sqrt{2 (\frac{x (x (- 9 + x)) + x (9 \cdot 3)}{3} + K)}$

Step 3.4.7.1.3

Factor $x$ out of $x (x (- 9 + x)) + x (9 \cdot 3)$ .

$y = \pm \sqrt{2 (\frac{x (x (- 9 + x) + 9 \cdot 3)}{3} + K)}$

Step 3.4.7.2

Apply the distributive property.

$y = \pm \sqrt{2 (\frac{x (x \cdot - 9 + x \cdot x + 9 \cdot 3)}{3} + K)}$

Step 3.4.7.3

Move $- 9$ to the left of $x$ .

$y = \pm \sqrt{2 (\frac{x (- 9 \cdot x + x \cdot x + 9 \cdot 3)}{3} + K)}$

Step 3.4.7.4

Multiply $x$ by $x$ .

$y = \pm \sqrt{2 (\frac{x (- 9 \cdot x + x^{2} + 9 \cdot 3)}{3} + K)}$

Step 3.4.7.5

Multiply $9$ by $3$ .

$y = \pm \sqrt{2 (\frac{x (- 9 x + x^{2} + 27)}{3} + K)}$

Step 3.4.7.6

Reorder terms.

$y = \pm \sqrt{2 (\frac{x (x^{2} - 9 x + 27)}{3} + K)}$

Step 3.4.8

To write $K$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{2 (\frac{x (x^{2} - 9 x + 27)}{3} + K \cdot \frac{3}{3})}$

Step 3.4.9

Simplify terms.

Tap for more steps...

Step 3.4.9.1

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

$y = \pm \sqrt{2 (\frac{x (x^{2} - 9 x + 27)}{3} + \frac{K \cdot 3}{3})}$

Step 3.4.9.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{2 \frac{x (x^{2} - 9 x + 27) + K \cdot 3}{3}}$

Step 3.4.10

Simplify the numerator.

Tap for more steps...

Step 3.4.10.1

Apply the distributive property.

$y = \pm \sqrt{2 \frac{x \cdot x^{2} + x (- 9 x) + x \cdot 27 + K \cdot 3}{3}}$

Step 3.4.10.2

Simplify.

Tap for more steps...

Step 3.4.10.2.1

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

Tap for more steps...

Step 3.4.10.2.1.1

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

Tap for more steps...

Step 3.4.10.2.1.1.1

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

$y = \pm \sqrt{2 \frac{x^{1} x^{2} + x (- 9 x) + x \cdot 27 + K \cdot 3}{3}}$

Step 3.4.10.2.1.1.2

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

$y = \pm \sqrt{2 \frac{x^{1 + 2} + x (- 9 x) + x \cdot 27 + K \cdot 3}{3}}$

Step 3.4.10.2.1.2

Add $1$ and $2$ .

$y = \pm \sqrt{2 \frac{x^{3} + x (- 9 x) + x \cdot 27 + K \cdot 3}{3}}$

Step 3.4.10.2.2

Rewrite using the commutative property of multiplication.

$y = \pm \sqrt{2 \frac{x^{3} - 9 x \cdot x + x \cdot 27 + K \cdot 3}{3}}$

Step 3.4.10.2.3

Move $27$ to the left of $x$ .

$y = \pm \sqrt{2 \frac{x^{3} - 9 x \cdot x + 27 \cdot x + K \cdot 3}{3}}$

Step 3.4.10.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.4.10.3.1

Move $x$ .

$y = \pm \sqrt{2 \frac{x^{3} - 9 (x \cdot x) + 27 \cdot x + K \cdot 3}{3}}$

Step 3.4.10.3.2

Multiply $x$ by $x$ .

$y = \pm \sqrt{2 \frac{x^{3} - 9 x^{2} + 27 \cdot x + K \cdot 3}{3}}$

$y = \pm \sqrt{2 \frac{x^{3} - 9 x^{2} + 27 x + K \cdot 3}{3}}$

Step 3.4.10.4

Move $3$ to the left of $K$ .

$y = \pm \sqrt{2 \frac{x^{3} - 9 x^{2} + 27 x + 3 K}{3}}$

Step 3.4.11

Combine $2$ and $\frac{x^{3} - 9 x^{2} + 27 x + 3 K}{3}$ .

$y = \pm \sqrt{\frac{2 (x^{3} - 9 x^{2} + 27 x + 3 K)}{3}}$

Step 3.4.12

Rewrite $\sqrt{\frac{2 (x^{3} - 9 x^{2} + 27 x + 3 K)}{3}}$ as $\frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{\sqrt{3}}$

Step 3.4.13

Multiply $\frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.4.14

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.14.1

Multiply $\frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.14.2

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

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.14.3

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

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.14.4

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

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.14.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.4.14.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 3.4.14.6.1

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

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.4.14.6.2

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

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.14.6.3

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

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.14.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.14.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.14.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{3^{1}}$

Step 3.4.14.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K)} \sqrt{3}}{3}$

Step 3.4.15

Simplify the numerator.

Tap for more steps...

Step 3.4.15.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{2 (x^{3} - 9 x^{2} + 27 x + 3 K) \cdot 3}}{3}$

Step 3.4.15.2

Multiply $3$ by $2$ .

$y = \pm \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$y = \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

Step 3.5.3

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

$y = \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

$y = - \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

$y = \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

$y = - \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

$y = \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

$y = - \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + 3 K)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + K)}}{3}$

$y = - \frac{\sqrt{6 (x^{3} - 9 x^{2} + 27 x + K)}}{3}$