Calculus Examples

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

Step 1

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

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Step 1.1

Rewrite.

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

Step 2

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

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Step 2.1

Differentiate $M$ with respect to $y$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$\frac{\partial M}{\partial y} = x \cdot 1 + \frac{d}{d y} [x^{2}]$

Step 2.3.3

Multiply $x$ by $1$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

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

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

Step 2.4.2

Add $x$ and $0$ .

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

Step 3

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

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Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

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

$\frac{\partial N}{\partial x} = 2 x$

Step 4

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

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Step 4.1

Substitute $x$ for $\frac{\partial M}{\partial y}$ and $2 x$ for $\frac{\partial N}{\partial x}$ .

$x = 2 x$

Step 4.2

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

$x = 2 x$ is not an identity.

Step 5

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

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Step 5.1

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

$\frac{x - \frac{\partial N}{\partial x}}{N}$

Step 5.2

Substitute $2 x$ for $\frac{\partial N}{\partial x}$ .

$\frac{x - (2 x)}{N}$

Step 5.3

Substitute $x^{2}$ for $N$ .

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Step 5.3.1

Substitute $x^{2}$ for $N$ .

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

Step 5.3.2

Simplify the numerator.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

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

$\frac{x^{1} - 1 \cdot 2 x}{x^{2}}$

Step 5.3.2.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - 1 \cdot 2 x}{x^{2}}$

Step 5.3.2.1.3

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

$\frac{x \cdot 1 + x (- 1 \cdot 2)}{x^{2}}$

Step 5.3.2.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \cdot 2)$ .

$\frac{x (1 - 1 \cdot 2)}{x^{2}}$

Step 5.3.2.2

Multiply $- 1$ by $2$ .

$\frac{x (1 - 2)}{x^{2}}$

Step 5.3.2.3

Subtract $2$ from $1$ .

$\frac{x \cdot - 1}{x^{2}}$

Step 5.3.3

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

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Step 5.3.3.1

Factor $x$ out of $x \cdot - 1$ .

$\frac{x (- 1)}{x^{2}}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

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

$\frac{x (- 1)}{x \cdot x}$

Step 5.3.3.2.2

Cancel the common factor.

$\frac{x \cdot - 1}{x \cdot x}$

Step 5.3.3.2.3

Rewrite the expression.

$\frac{- 1}{x}$

Step 5.3.4

Move the negative in front of the fraction.

$- \frac{1}{x}$

Step 5.4

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

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

Step 6

Evaluate the integral $e^{\int - \frac{1}{x} d x}$ .

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Step 6.1

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

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

Step 6.2

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

$μ (x, y) = e^{- (\ln (| x |) + C)}$

Step 6.3

Simplify.

$μ (x, y) = e^{- \ln (| x |)} + C$

Step 6.4

Simplify each term.

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Step 6.4.1

Simplify $- \ln (| x |)$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x |}^{- 1})} + C$

Step 6.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{- 1} + C$

Step 6.4.3

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

$μ (x, y) = \frac{1}{| x |} + C$

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

Factor $x$ out of $x y$ .

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

Step 7.3.2

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

$\frac{x (y) + x \cdot x}{x} d x + x^{2} d y = 0$

Step 7.3.3

Factor $x$ out of $x (y) + x \cdot x$ .

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

Step 7.4

Cancel the common factor of $x$ .

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Divide $y + x$ by $1$ .

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

Step 7.5

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

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

Step 7.6

Cancel the common factor of $x$ .

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Step 7.6.1

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

$(y + x) d x + x \cdot x \frac{1}{x} d y = 0$

Step 7.6.2

Cancel the common factor.

$(y + x) d x + x \cdot x \frac{1}{x} d y = 0$

Step 7.6.3

Rewrite the expression.

$(y + x) d x + x d y = 0$

Step 8

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

$f (x, y) = \int x d y$

Step 9

Integrate $N (x, y) = x$ to find $f (x, y)$ .

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Step 9.1

Apply the constant rule.

$f (x, y) = x y + C$

Step 10

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

$f (x, y) = x y + g (x)$

Step 11

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

$\frac{\partial f}{\partial x} = y + x$

Step 12

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

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Step 12.1

Differentiate $f$ with respect to $x$ .

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

Step 12.2

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

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

Multiply $y$ by $1$ .

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

Step 12.4

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

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

Step 12.5

Reorder terms.

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

Step 13

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

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Step 13.1

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

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Step 13.1.1

Subtract $y$ from both sides of the equation.

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

Step 13.1.2

Combine the opposite terms in $y + x - y$ .

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Step 13.1.2.1

Subtract $y$ from $y$ .

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

Step 13.1.2.2

Add $x$ and $0$ .

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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 15

Substitute for $g (x)$ in $f (x, y) = x y + g (x)$ .

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

Step 16

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

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