Calculus Examples

$(x - y^{3} + y^{2} \sin (x)) d x = (3 x y^{2} + 2 y \cos (x)) d y$

Step 1

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

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Step 1.1

Subtract $(3 x y^{2} + 2 y \cos (x)) d y$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Differentiate $M$ with respect to $y$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [- y^{3}] + \frac{d}{d y} [y^{2} \sin (x)]$

Step 2.3

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

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Step 2.3.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y^{3}$ with respect to $y$ is $- \frac{d}{d y} [y^{3}]$ .

$\frac{\partial M}{\partial y} = 0 - \frac{d}{d y} [y^{3}] + \frac{d}{d y} [y^{2} \sin (x)]$

Step 2.3.2

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

$\frac{\partial M}{\partial y} = 0 - (3 y^{2}) + \frac{d}{d y} [y^{2} \sin (x)]$

Step 2.3.3

Multiply $3$ by $- 1$ .

$\frac{\partial M}{\partial y} = 0 - 3 y^{2} + \frac{d}{d y} [y^{2} \sin (x)]$

Step 2.4

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

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Step 2.4.1

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

$\frac{\partial M}{\partial y} = 0 - 3 y^{2} + \sin (x) \frac{d}{d y} [y^{2}]$

Step 2.4.2

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

$\frac{\partial M}{\partial y} = 0 - 3 y^{2} + \sin (x) (2 y)$

Step 2.4.3

Move $2$ to the left of $\sin (x)$ .

$\frac{\partial M}{\partial y} = 0 - 3 y^{2} + 2 \sin (x) y$

Step 2.5

Simplify.

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Step 2.5.1

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

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

Step 2.5.2

Reorder terms.

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

Step 3

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

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Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (3 x y^{2} + 2 y \cos (x))$ with respect to $x$ is $- \frac{d}{d x} [3 x y^{2} + 2 y \cos (x)]$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

$\frac{\partial N}{\partial x} = - (3 y^{2} \cdot 1 + \frac{d}{d x} [2 y \cos (x)])$

Step 3.2.5

Multiply $3$ by $1$ .

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

Step 3.2.6

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

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

Step 3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\frac{\partial N}{\partial x} = - (3 y^{2} + 2 y (- \sin (x)))$

Step 3.4

Multiply $- 1$ by $2$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Multiply $3$ by $- 1$ .

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

Step 3.5.2.2

Multiply $- 2$ by $- 1$ .

$\frac{\partial N}{\partial x} = - 3 y^{2} + 2 y \sin (x)$

Step 4

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

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Step 4.1

Substitute $- 3 y^{2} + 2 y \sin (x)$ for $\frac{\partial M}{\partial y}$ and $- 3 y^{2} + 2 y \sin (x)$ for $\frac{\partial N}{\partial x}$ .

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

Step 4.2

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

$- 3 y^{2} + 2 y \sin (x) = - 3 y^{2} + 2 y \sin (x)$ is an identity.

Step 5

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

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

Step 6

Integrate $N (x, y) = - (3 x y^{2} + 2 y \cos (x))$ to find $f (x, y)$ .

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Step 6.1

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

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

Step 6.2

Split the single integral into multiple integrals.

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Since $2 \cos (x)$ is constant with respect to $y$ , move $2 \cos (x)$ out of the integral.

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

Step 6.6

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

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

Step 6.7

Simplify.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

Differentiate $f$ with respect to $x$ .

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

Step 9.2

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

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

Step 9.3

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

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Step 9.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x y^{3} + \cos (x) y^{2})$ with respect to $x$ is $- \frac{d}{d x} [x y^{3} + \cos (x) y^{2}]$ .

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

Step 9.3.2

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

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

Step 9.3.3

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

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

Step 9.3.4

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

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

Step 9.3.5

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

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

Step 9.3.6

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 9.3.7

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

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

Step 9.4

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

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

Step 9.5

Simplify.

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Step 9.5.1

Apply the distributive property.

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

Step 9.5.2

Combine terms.

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Step 9.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 9.5.2.2

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

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

Step 9.5.3

Reorder terms.

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

Step 10

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

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Step 10.1

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

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Step 10.1.1

Add $y^{3}$ to both sides of the equation.

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

Step 10.1.2

Subtract $y^{2} \sin (x)$ from both sides of the equation.

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

Step 10.1.3

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

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Step 10.1.3.1

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

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

Step 10.1.3.2

Add $x + y^{2} \sin (x)$ and $0$ .

$\frac{d g}{d x} = x + y^{2} \sin (x) - y^{2} \sin (x)$

Step 10.1.3.3

Subtract $y^{2} \sin (x)$ from $y^{2} \sin (x)$ .

$\frac{d g}{d x} = x + 0$

Step 10.1.3.4

Add $x$ and $0$ .

$\frac{d g}{d x} = x$

Step 11

Find the antiderivative of $x$ to find $g (x)$ .

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Step 11.1

Integrate both sides of $\frac{d g}{d x} = x$ .

$\int \frac{d g}{d x} d x = \int x d x$

Step 11.2

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

$g (x) = \int x d x$

Step 11.3

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

$g (x) = \frac{1}{2} x^{2} + C$

Step 12

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

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

Step 13

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

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the distributive property.

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

Step 13.1.2

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

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

Step 13.2

Reorder factors in $f (x, y) = - x y^{3} - \cos (x) y^{2} + \frac{x^{2}}{2} + C$ .

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