Calculus Examples

$(2 x^{2} + 3 y^{2}) d y + (3 x^{2} + 4 x y) d x = 0$

Step 1

Rewrite the differential equation to fit the Exact differential equation technique.

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Step 1.1

Rewrite.

$(3 x^{2} + 4 x y) d x + (2 x^{2} + 3 y^{2}) d y = 0$

Step 2

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 3 x^{2} + 4 x y$ .

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Step 2.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 x^{2} + 4 x y]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $3 x^{2} + 4 x y$ with respect to $y$ is $\frac{d}{d y} [3 x^{2}] + \frac{d}{d y} [4 x y]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 x^{2}] + \frac{d}{d y} [4 x y]$

Step 2.2.2

Since $3 x^{2}$ is constant with respect to $y$ , the derivative of $3 x^{2}$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [4 x y]$

Step 2.3

Evaluate $\frac{d}{d y} [4 x y]$ .

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Step 2.3.1

Since $4 x$ is constant with respect to $y$ , the derivative of $4 x y$ with respect to $y$ is $4 x \frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = 0 + 4 x \frac{d}{d y} [y]$

Step 2.3.2

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

$\frac{\partial M}{\partial y} = 0 + 4 x \cdot 1$

Step 2.3.3

Multiply $4$ by $1$ .

$\frac{\partial M}{\partial y} = 0 + 4 x$

Step 2.4

Add $0$ and $4 x$ .

$\frac{\partial M}{\partial y} = 4 x$

Step 3

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

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Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $2$ by $2$ .

$\frac{\partial N}{\partial x} = 4 x + \frac{d}{d x} [3 y^{2}]$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $3 y^{2}$ is constant with respect to $x$ , the derivative of $3 y^{2}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 4 x + 0$

Step 3.4.2

Add $4 x$ and $0$ .

$\frac{\partial N}{\partial x} = 4 x$

Step 4

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

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Step 4.1

Substitute $4 x$ for $\frac{\partial M}{\partial y}$ and $4 x$ for $\frac{\partial N}{\partial x}$ .

$4 x = 4 x$

Step 4.2

Since the two sides have been shown to be equivalent, the equation is an identity.

$4 x = 4 x$ is an identity.

Step 5

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

$f (x, y) = \int 3 x^{2} + 4 x y d x$

Step 6

Integrate $M (x, y) = 3 x^{2} + 4 x y$ to find $f (x, y)$ .

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Step 6.1

Split the single integral into multiple integrals.

$f (x, y) = \int 3 x^{2} d x + \int 4 x y d x$

Step 6.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$f (x, y) = 3 \int x^{2} d x + \int 4 x y d x$

Step 6.3

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

$f (x, y) = 3 (\frac{1}{3} x^{3} + C) + \int 4 x y d x$

Step 6.4

Since $4 y$ is constant with respect to $x$ , move $4 y$ out of the integral.

$f (x, y) = 3 (\frac{1}{3} x^{3} + C) + 4 y \int x d x$

Step 6.5

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

$f (x, y) = 3 (\frac{1}{3} x^{3} + C) + 4 y (\frac{1}{2} x^{2} + C)$

Step 6.6

Simplify.

$f (x, y) = 3 (\frac{1}{3}) x^{3} + 4 y \frac{1}{2} x^{2} + C$

Step 6.7

Simplify.

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Step 6.7.1

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

$f (x, y) = \frac{3}{3} x^{3} + 4 y \frac{1}{2} x^{2} + C$

Step 6.7.2

Cancel the common factor of $3$ .

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Step 6.7.2.1

Cancel the common factor.

$f (x, y) = \frac{3}{3} x^{3} + 4 y \frac{1}{2} x^{2} + C$

Step 6.7.2.2

Rewrite the expression.

$f (x, y) = 1 x^{3} + 4 y \frac{1}{2} x^{2} + C$

Step 6.7.3

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

$f (x, y) = x^{3} + 4 y \frac{1}{2} x^{2} + C$

Step 6.7.4

Combine $4$ and $\frac{1}{2}$ .

$f (x, y) = x^{3} + y \frac{4}{2} x^{2} + C$

Step 6.7.5

Combine $y$ and $\frac{4}{2}$ .

$f (x, y) = x^{3} + \frac{y \cdot 4}{2} x^{2} + C$

Step 6.7.6

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

$f (x, y) = x^{3} + \frac{4 \cdot y}{2} x^{2} + C$

Step 6.7.7

Multiply $4$ by $y$ .

$f (x, y) = x^{3} + \frac{4 y}{2} x^{2} + C$

Step 6.7.8

Cancel the common factor of $4$ and $2$ .

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Step 6.7.8.1

Factor $2$ out of $4 y$ .

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

Step 6.7.8.2

Cancel the common factors.

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Step 6.7.8.2.1

Factor $2$ out of $2$ .

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

Step 6.7.8.2.2

Cancel the common factor.

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

Step 6.7.8.2.3

Rewrite the expression.

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

Step 6.7.8.2.4

Divide $2 y$ by $1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

Differentiate $f$ with respect to $y$ .

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

Step 9.2

Differentiate.

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Step 9.2.1

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

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

Step 9.2.2

Since $x^{3}$ is constant with respect to $y$ , the derivative of $x^{3}$ with respect to $y$ is $0$ .

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.3.3

Multiply $2$ by $1$ .

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

Step 9.4

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

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

Step 9.5

Simplify.

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Step 9.5.1

Add $0$ and $2 x^{2}$ .

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

Step 9.5.2

Reorder terms.

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

Step 10

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

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Step 10.1

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

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Step 10.1.1

Subtract $2 x^{2}$ from both sides of the equation.

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

Step 10.1.2

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

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Step 10.1.2.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

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

Step 10.1.2.2

Add $3 y^{2}$ and $0$ .

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

Step 11

Find the antiderivative of $3 y^{2}$ to find $g (y)$ .

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Step 11.1

Integrate both sides of $\frac{d g}{d y} = 3 y^{2}$ .

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

Step 11.2

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

$g (y) = \int 3 y^{2} d y$

Step 11.3

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$g (y) = 3 \int y^{2} d y$

Step 11.4

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

$g (y) = 3 (\frac{1}{3} y^{3} + C)$

Step 11.5

Simplify the answer.

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Step 11.5.1

Rewrite $3 (\frac{1}{3} y^{3} + C)$ as $3 (\frac{1}{3}) y^{3} + C$ .

$g (y) = 3 (\frac{1}{3}) y^{3} + C$

Step 11.5.2

Simplify.

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Step 11.5.2.1

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

$g (y) = \frac{3}{3} y^{3} + C$

Step 11.5.2.2

Cancel the common factor of $3$ .

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Step 11.5.2.2.1

Cancel the common factor.

$g (y) = \frac{3}{3} y^{3} + C$

Step 11.5.2.2.2

Rewrite the expression.

$g (y) = 1 y^{3} + C$

Step 11.5.2.3

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

$g (y) = y^{3} + C$

Step 12

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

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