Calculus Examples

$(2 x y^{2} - 3 y^{3}) d x + (7 - 3 x y^{2}) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 2 x y^{2} - 3 y^{3}$ .

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [2 x y^{2} - 3 y^{3}]$

Step 1.2

By the Sum Rule, the derivative of $2 x y^{2} - 3 y^{3}$ with respect to $y$ is $\frac{d}{d y} [2 x y^{2}] + \frac{d}{d y} [- 3 y^{3}]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [2 x y^{2}] + \frac{d}{d y} [- 3 y^{3}]$

Step 1.3

Evaluate $\frac{d}{d y} [2 x y^{2}]$ .

Tap for more steps...

Step 1.3.1

Since $2 x$ is constant with respect to $y$ , the derivative of $2 x y^{2}$ with respect to $y$ is $2 x \frac{d}{d y} [y^{2}]$ .

$\frac{\partial M}{\partial y} = 2 x \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 3 y^{3}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$\frac{\partial M}{\partial y} = 2 x (2 y) + \frac{d}{d y} [- 3 y^{3}]$

Step 1.3.3

Multiply $2$ by $2$ .

$\frac{\partial M}{\partial y} = 4 x y + \frac{d}{d y} [- 3 y^{3}]$

Step 1.4

Evaluate $\frac{d}{d y} [- 3 y^{3}]$ .

Tap for more steps...

Step 1.4.1

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3 y^{3}$ with respect to $y$ is $- 3 \frac{d}{d y} [y^{3}]$ .

$\frac{\partial M}{\partial y} = 4 x y - 3 \frac{d}{d y} [y^{3}]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$\frac{\partial M}{\partial y} = 4 x y - 3 (3 y^{2})$

Step 1.4.3

Multiply $3$ by $- 3$ .

$\frac{\partial M}{\partial y} = 4 x y - 9 y^{2}$

Step 1.5

Reorder terms.

$\frac{\partial M}{\partial y} = - 9 y^{2} + 4 x y$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 7 - 3 x y^{2}$ .

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [7 - 3 x y^{2}]$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

By the Sum Rule, the derivative of $7 - 3 x y^{2}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 3 x y^{2}]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [7] + \frac{d}{d x} [- 3 x y^{2}]$

Step 2.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 0 + \frac{d}{d x} [- 3 x y^{2}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 3 x y^{2}]$ .

Tap for more steps...

Step 2.3.1

Since $- 3 y^{2}$ is constant with respect to $x$ , the derivative of $- 3 x y^{2}$ with respect to $x$ is $- 3 y^{2} \frac{d}{d x} [x]$ .

$\frac{\partial N}{\partial x} = 0 - 3 y^{2} \frac{d}{d x} [x]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{\partial N}{\partial x} = 0 - 3 y^{2} \cdot 1$

Step 2.3.3

Multiply $- 3$ by $1$ .

$\frac{\partial N}{\partial x} = 0 - 3 y^{2}$

Step 2.4

Subtract $3 y^{2}$ from $0$ .

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

Step 3

Check that $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

Tap for more steps...

Step 3.1

Substitute $- 9 y^{2} + 4 x y$ for $\frac{\partial M}{\partial y}$ and $- 3 y^{2}$ for $\frac{\partial N}{\partial x}$ .

$- 9 y^{2} + 4 x y = - 3 y^{2}$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$- 9 y^{2} + 4 x y = - 3 y^{2}$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

Tap for more steps...

Step 4.1

Substitute $- 3 y^{2}$ for $\frac{\partial N}{\partial x}$ .

$\frac{- 3 y^{2} - \frac{\partial M}{\partial y}}{M}$

Step 4.2

Substitute $- 9 y^{2} + 4 x y$ for $\frac{\partial M}{\partial y}$ .

$\frac{- 3 y^{2} - (- 9 y^{2} + 4 x y)}{M}$

Step 4.3

Substitute $2 x y^{2} - 3 y^{3}$ for $M$ .

Tap for more steps...

Step 4.3.1

Substitute $2 x y^{2} - 3 y^{3}$ for $M$ .

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

Step 4.3.2

Simplify the numerator.

Tap for more steps...

Step 4.3.2.1

Apply the distributive property.

$\frac{- 3 y^{2} - (- 9 y^{2}) - (4 x y)}{2 x y^{2} - 3 y^{3}}$

Step 4.3.2.2

Multiply $- 9$ by $- 1$ .

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

Step 4.3.2.3

Multiply $4$ by $- 1$ .

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

Step 4.3.2.4

Add $- 3 y^{2}$ and $9 y^{2}$ .

$\frac{6 y^{2} - 4 x y}{2 x y^{2} - 3 y^{3}}$

Step 4.3.2.5

Factor $2 y$ out of $6 y^{2} - 4 x y$ .

Tap for more steps...

Step 4.3.2.5.1

Factor $2 y$ out of $6 y^{2}$ .

$\frac{2 y (3 y) - 4 x y}{2 x y^{2} - 3 y^{3}}$

Step 4.3.2.5.2

Factor $2 y$ out of $- 4 x y$ .

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

Step 4.3.2.5.3

Factor $2 y$ out of $2 y (3 y) + 2 y (- 2 x)$ .

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

Step 4.3.3

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

Tap for more steps...

Step 4.3.3.1

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

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

Step 4.3.3.2

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

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

Step 4.3.3.3

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

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

Step 4.3.4

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

Tap for more steps...

Step 4.3.4.1

Factor $y$ out of $2 y (3 y - 2 x)$ .

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

Step 4.3.4.2

Cancel the common factors.

Tap for more steps...

Step 4.3.4.2.1

Factor $y$ out of $y^{2} (2 x - 3 y)$ .

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

Step 4.3.4.2.2

Cancel the common factor.

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

Step 4.3.4.2.3

Rewrite the expression.

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

Step 4.3.5

Cancel the common factor of $3 y - 2 x$ and $2 x - 3 y$ .

Tap for more steps...

Step 4.3.5.1

Factor $- 1$ out of $3 y$ .

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

Step 4.3.5.2

Factor $- 1$ out of $- 2 x$ .

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

Step 4.3.5.3

Factor $- 1$ out of $- (- 3 y) - (2 x)$ .

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

Step 4.3.5.4

Rewrite $- (- 3 y + 2 x)$ as $- 1 (- 3 y + 2 x)$ .

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

Step 4.3.5.5

Reorder terms.

$\frac{2 (- 1 (2 x - 3 y))}{y (2 x - 3 y)}$

Step 4.3.5.6

Cancel the common factor.

$\frac{2 (- 1 (2 x - 3 y))}{y (2 x - 3 y)}$

Step 4.3.5.7

Rewrite the expression.

$\frac{2 \cdot (- 1)}{y}$

Step 4.3.6

Multiply $2$ by $- 1$ .

$\frac{- 2}{y}$

Step 4.3.7

Substitute $2 x y^{2} - 3 y^{3}$ for $M$ .

$- \frac{2}{y}$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

$μ (x, y) = e^{\int - \frac{2}{y} d y}$

Step 5

Evaluate the integral $e^{\int - \frac{2}{y} d y}$ .

Tap for more steps...

Step 5.1

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

$μ (x, y) = e^{- \int \frac{2}{y} d y}$

Step 5.2

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

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

Step 5.3

Multiply $2$ by $- 1$ .

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

Step 5.4

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

$μ (x, y) = e^{- 2 (\ln (| y |) + C)}$

Step 5.5

Simplify.

$μ (x, y) = e^{- 2 \ln (| y |)} + C$

Step 5.6

Simplify each term.

Tap for more steps...

Step 5.6.1

Simplify $- 2 \ln (| y |)$ by moving $- 2$ inside the logarithm.

$μ (x, y) = e^{\ln ({| y |}^{- 2})} + C$

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{- 2} + C$

Step 5.6.3

Remove the absolute value in ${| y |}^{- 2}$ because exponentiations with even powers are always positive.

$μ (x, y) = y^{- 2} + C$

Step 5.6.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$μ (x, y) = \frac{1}{y^{2}} + C$

Step 6

Multiply both sides of $(2 x y^{2} - 3 y^{3}) d x + (7 - 3 x y^{2}) d y = 0$ by the integration factor $\frac{1}{y^{2}}$ .

Tap for more steps...

Step 6.1

Multiply $2 x y^{2} - 3 y^{3}$ by $\frac{1}{y^{2}}$ .

$(2 x y^{2} - 3 y^{3}) \frac{1}{y^{2}} d x + (7 - 3 x y^{2}) d y = 0$

Step 6.2

Multiply $2 x y^{2} - 3 y^{3}$ by $\frac{1}{y^{2}}$ .

$\frac{2 x y^{2} - 3 y^{3}}{y^{2}} d x + (7 - 3 x y^{2}) d y = 0$

Step 6.3

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

Tap for more steps...

Step 6.3.1

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

$\frac{y^{2} (2 x) - 3 y^{3}}{y^{2}} d x + (7 - 3 x y^{2}) d y = 0$

Step 6.3.2

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

$\frac{y^{2} (2 x) + y^{2} (- 3 y)}{y^{2}} d x + (7 - 3 x y^{2}) d y = 0$

Step 6.3.3

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

$\frac{y^{2} (2 x - 3 y)}{y^{2}} d x + (7 - 3 x y^{2}) d y = 0$

Step 6.4

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

Tap for more steps...

Step 6.4.1

Cancel the common factor.

$\frac{y^{2} (2 x - 3 y)}{y^{2}} d x + (7 - 3 x y^{2}) d y = 0$

Step 6.4.2

Divide $2 x - 3 y$ by $1$ .

$(2 x - 3 y) d x + (7 - 3 x y^{2}) d y = 0$

Step 6.5

Multiply $7 - 3 x y^{2}$ by $\frac{1}{y^{2}}$ .

$(2 x - 3 y) d x + (7 - 3 x y^{2}) \frac{1}{y^{2}} d y = 0$

Step 6.6

Multiply $7 - 3 x y^{2}$ by $\frac{1}{y^{2}}$ .

$(2 x - 3 y) d x + \frac{7 - 3 x y^{2}}{y^{2}} d y = 0$

Step 7

Set $f (x, y)$ equal to the integral of $M (x, y)$ .

$f (x, y) = \int 2 x - 3 y d x$

Step 8

Integrate $M (x, y) = 2 x - 3 y$ to find $f (x, y)$ .

Tap for more steps...

Step 8.1

Split the single integral into multiple integrals.

$f (x, y) = \int 2 x d x + \int - 3 y d x$

Step 8.2

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

$f (x, y) = 2 \int x d x + \int - 3 y d x$

Step 8.3

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

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

Step 8.4

Apply the constant rule.

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

Step 8.5

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

$f (x, y) = 2 (\frac{x^{2}}{2} + C) - 3 y x + C$

Step 8.6

Simplify.

$f (x, y) = x^{2} - 3 y x + C$

Step 9

Since the integral of $g (y)$ will contain an integration constant, we can replace $C$ with $g (y)$ .

$f (x, y) = x^{2} - 3 y x + g (y)$

Step 10

Set $\frac{\partial f}{\partial y} = N (x, y)$ .

$\frac{\partial f}{\partial y} = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11

Find $\frac{\partial f}{\partial y}$ .

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [x^{2} - 3 y x + g (y)] = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.2

Differentiate.

Tap for more steps...

Step 11.2.1

By the Sum Rule, the derivative of $x^{2} - 3 y x + g (y)$ with respect to $y$ is $\frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 3 y x] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 3 y x] + \frac{d}{d y} [g (y)] = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.2.2

Since $x^{2}$ is constant with respect to $y$ , the derivative of $x^{2}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- 3 y x] + \frac{d}{d y} [g (y)] = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.3

Evaluate $\frac{d}{d y} [- 3 y x]$ .

Tap for more steps...

Step 11.3.1

Since $- 3 x$ is constant with respect to $y$ , the derivative of $- 3 y x$ with respect to $y$ is $- 3 x \frac{d}{d y} [y]$ .

$0 - 3 x \frac{d}{d y} [y] + \frac{d}{d y} [g (y)] = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$0 - 3 x \cdot 1 + \frac{d}{d y} [g (y)] = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.3.3

Multiply $- 3$ by $1$ .

$0 - 3 x + \frac{d}{d y} [g (y)] = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.4

Differentiate using the function rule which states that the derivative of $g (y)$ is $\frac{d g}{d y}$ .

$0 - 3 x + \frac{d g}{d y} = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.5

Simplify.

Tap for more steps...

Step 11.5.1

Subtract $3 x$ from $0$ .

$- 3 x + \frac{d g}{d y} = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 11.5.2

Reorder terms.

$\frac{d g}{d y} - 3 x = \frac{7 - 3 x y^{2}}{y^{2}}$

Step 12

Solve for $\frac{d g}{d y}$ .

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

Add $3 x$ to both sides of the equation.

$\frac{d g}{d y} = \frac{7 - 3 x y^{2}}{y^{2}} + 3 x$

Step 12.1.2

Simplify each term.

Tap for more steps...

Step 12.1.2.1

Split the fraction $\frac{7 - 3 x y^{2}}{y^{2}}$ into two fractions.

$\frac{d g}{d y} = \frac{7}{y^{2}} + \frac{- 3 x y^{2}}{y^{2}} + 3 x$

Step 12.1.2.2

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

Tap for more steps...

Step 12.1.2.2.1

Cancel the common factor.

$\frac{d g}{d y} = \frac{7}{y^{2}} + \frac{- 3 x y^{2}}{y^{2}} + 3 x$

Step 12.1.2.2.2

Divide $- 3 x$ by $1$ .

$\frac{d g}{d y} = \frac{7}{y^{2}} - 3 x + 3 x$

Step 12.1.3

Combine the opposite terms in $\frac{7}{y^{2}} - 3 x + 3 x$ .

Tap for more steps...

Step 12.1.3.1

Add $- 3 x$ and $3 x$ .

$\frac{d g}{d y} = \frac{7}{y^{2}} + 0$

Step 12.1.3.2

Add $\frac{7}{y^{2}}$ and $0$ .

$\frac{d g}{d y} = \frac{7}{y^{2}}$

Step 13

Find the antiderivative of $\frac{7}{y^{2}}$ to find $g (y)$ .

Tap for more steps...

Step 13.1

Integrate both sides of $\frac{d g}{d y} = \frac{7}{y^{2}}$ .

$\int \frac{d g}{d y} d y = \int \frac{7}{y^{2}} d y$

Step 13.2

Evaluate $\int \frac{d g}{d y} d y$ .

$g (y) = \int \frac{7}{y^{2}} d y$

Step 13.3

Since $7$ is constant with respect to $y$ , move $7$ out of the integral.

$g (y) = 7 \int \frac{1}{y^{2}} d y$

Step 13.4

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

$g (y) = 7 \int {(y^{2})}^{- 1} d y$

Step 13.5

Multiply the exponents in ${(y^{2})}^{- 1}$ .

Tap for more steps...

Step 13.5.1

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

$g (y) = 7 \int y^{2 \cdot - 1} d y$

Step 13.5.2

Multiply $2$ by $- 1$ .

$g (y) = 7 \int y^{- 2} d y$

Step 13.6

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

$g (y) = 7 (- y^{- 1} + C)$

Step 13.7

Simplify the answer.

Tap for more steps...

Step 13.7.1

Rewrite $7 (- y^{- 1} + C)$ as $7 (- \frac{1}{y}) + C$ .

$g (y) = 7 (- \frac{1}{y}) + C$

Step 13.7.2

Simplify.

Tap for more steps...

Step 13.7.2.1

Multiply $- 1$ by $7$ .

$g (y) = - 7 \frac{1}{y} + C$

Step 13.7.2.2

Combine $- 7$ and $\frac{1}{y}$ .

$g (y) = \frac{- 7}{y} + C$

Step 13.7.2.3

Move the negative in front of the fraction.

$g (y) = - \frac{7}{y} + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = x^{2} - 3 y x + g (y)$ .

$f (x, y) = x^{2} - 3 y x - \frac{7}{y} + C$