Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

$\frac{\partial M}{\partial y} = 0 + 2 y + \frac{d}{d y} [2 x]$

Step 1.5

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

$\frac{\partial M}{\partial y} = 0 + 2 y + 0$

Step 1.6

Combine terms.

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Step 1.6.1

Add $0$ and $2 y$ .

$\frac{\partial M}{\partial y} = 2 y + 0$

Step 1.6.2

Add $2 y$ and $0$ .

$\frac{\partial M}{\partial y} = 2 y$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 3

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

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Step 3.1

Substitute $2 y$ for $\frac{\partial M}{\partial y}$ and $0$ for $\frac{\partial N}{\partial x}$ .

$2 y = 0$

Step 3.2

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

$2 y = 0$ is not an identity.

Step 4

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 4.1

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

$\frac{2 y - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $0$ for $\frac{\partial N}{\partial x}$ .

$\frac{2 y + 0}{N}$

Step 4.3

Substitute $2 y$ for $N$ .

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Step 4.3.1

Substitute $2 y$ for $N$ .

$\frac{2 y + 0}{2 y}$

Step 4.3.2

Cancel the common factor of $2 y + 0$ and $2$ .

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Step 4.3.2.1

Factor $2$ out of $2 y$ .

$\frac{2 (y) + 0}{2 y}$

Step 4.3.2.2

Factor $2$ out of $0$ .

$\frac{2 (y) + 2 \cdot 0}{2 y}$

Step 4.3.2.3

Factor $2$ out of $2 (y) + 2 (0)$ .

$\frac{2 (y + 0)}{2 y}$

Step 4.3.2.4

Cancel the common factors.

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Step 4.3.2.4.1

Factor $2$ out of $2 y$ .

$\frac{2 (y + 0)}{2 (y)}$

Step 4.3.2.4.2

Cancel the common factor.

$\frac{2 (y + 0)}{2 y}$

Step 4.3.2.4.3

Rewrite the expression.

$\frac{y + 0}{y}$

Step 4.3.3

Add $y$ and $0$ .

$\frac{y}{y}$

Step 4.3.4

Cancel the common factor of $y$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{y}{y}$

Step 4.3.4.2

Rewrite the expression.

$1$

Step 4.4

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

$μ (x, y) = e^{\int d x}$

Step 5

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

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Step 5.1

Apply the constant rule.

$μ (x, y) = e^{x + C}$

Step 5.2

Simplify.

$μ (x, y) = e^{x} + C$

Step 6

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

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Step 6.1

Multiply $x^{2} + y^{2} + 2 x$ by $e^{x}$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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

Step 7

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

$f (x, y) = \int 2 y e^{x} d y$

Step 8

Integrate $N (x, y) = 2 y e^{x}$ to find $f (x, y)$ .

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Step 8.1

Since $2 e^{x}$ is constant with respect to $y$ , move $2 e^{x}$ out of the integral.

$f (x, y) = 2 e^{x} \int y d y$

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $2 e^{x} (\frac{1}{2} y^{2} + C)$ as $2 e^{x} \frac{1}{2} y^{2} + C$ .

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

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

Step 8.3.2.3

Cancel the common factor of $2$ .

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Step 8.3.2.3.1

Cancel the common factor.

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

Step 8.3.2.3.2

Divide $e^{x}$ by $1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 11.4

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

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

Step 11.5

Simplify.

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Step 11.5.1

Reorder terms.

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

Step 11.5.2

Reorder factors in $\frac{d g}{d x} + e^{x} y^{2}$ .

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

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

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

Step 12.1.2

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

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Step 12.1.2.1

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

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

Step 12.1.2.2

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

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

Step 13

Find the antiderivative of $x^{2} e^{x} + 2 x e^{x}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = x^{2} e^{x} + 2 x e^{x}$ .

$\int \frac{d g}{d x} d x = \int x^{2} e^{x} + 2 x e^{x} d x$

Step 13.2

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

$g (x) = \int x^{2} e^{x} + 2 x e^{x} d x$

Step 13.3

Split the single integral into multiple integrals.

$g (x) = \int x^{2} e^{x} d x + \int 2 x e^{x} d x$

Step 13.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{x}$ .

$g (x) = x^{2} e^{x} - \int e^{x} (2 x) d x + \int 2 x e^{x} d x$

Step 13.5

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

$g (x) = x^{2} e^{x} - (2 \int e^{x} (x) d x) + \int 2 x e^{x} d x$

Step 13.6

Multiply $2$ by $- 1$ .

$g (x) = x^{2} e^{x} - 2 \int e^{x} (x) d x + \int 2 x e^{x} d x$

Step 13.7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$g (x) = x^{2} e^{x} - 2 (x e^{x} - \int e^{x} d x) + \int 2 x e^{x} d x$

Step 13.8

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$g (x) = x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C)) + \int 2 x e^{x} d x$

Step 13.9

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

$g (x) = x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C)) + 2 \int x e^{x} d x$

Step 13.10

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$g (x) = x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C)) + 2 (x e^{x} - \int e^{x} d x)$

Step 13.11

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$g (x) = x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C)) + 2 (x e^{x} - (e^{x} + C))$

Step 13.12

Simplify.

$g (x) = x^{2} e^{x} - 2 x e^{x} + 2 e^{x} + 2 (x e^{x} - e^{x}) + C$

Step 14

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

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

Step 15

Simplify $f (x, y) = e^{x} y^{2} + x^{2} e^{x} - 2 x e^{x} + 2 e^{x} + 2 (x e^{x} - e^{x}) + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

Apply the distributive property.

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

Step 15.1.2

Multiply $- 1$ by $2$ .

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

Step 15.2

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

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Step 15.2.1

Add $- 2 x e^{x}$ and $2 x e^{x}$ .

$f (x, y) = e^{x} y^{2} + x^{2} e^{x} + 2 e^{x} + 0 - 2 e^{x} + C$

Step 15.2.2

Add $e^{x} y^{2} + x^{2} e^{x} + 2 e^{x}$ and $0$ .

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

Step 15.2.3

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

$f (x, y) = e^{x} y^{2} + x^{2} e^{x} + 0 + C$

Step 15.2.4

Add $e^{x} y^{2} + x^{2} e^{x}$ and $0$ .

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

Step 15.3

Reorder factors in $f (x, y) = e^{x} y^{2} + x^{2} e^{x} + C$ .

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