Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 3 x^{2} \cdot 1 + \frac{d}{d y} [2 x y] + \frac{d}{d y} [y^{3}]$

Step 1.3.3

Multiply $3$ by $1$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

$\frac{\partial M}{\partial y} = 3 x^{2} + 2 x \cdot 1 + \frac{d}{d y} [y^{3}]$

Step 1.4.3

Multiply $2$ by $1$ .

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

Step 1.5

Differentiate using the Power Rule.

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Step 1.5.1

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

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

Step 1.5.2

Reorder terms.

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$\frac{\partial N}{\partial x} = 2 x + 0$

Step 2.5

Add $2 x$ and $0$ .

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

Step 3

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

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Step 3.1

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

$3 x^{2} + 3 y^{2} + 2 x = 2 x$

Step 3.2

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

$3 x^{2} + 3 y^{2} + 2 x = 2 x$ 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 $3 x^{2} + 3 y^{2} + 2 x$ for $\frac{\partial M}{\partial y}$ .

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Multiply $- 1$ by $2$ .

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

Step 4.3.2.2

Subtract $2 x$ from $2 x$ .

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

Step 4.3.2.3

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

$\frac{3 x^{2} + 3 y^{2}}{x^{2} + y^{2}}$

Step 4.3.2.4

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

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Step 4.3.2.4.1

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

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

Step 4.3.2.4.2

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

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

Step 4.3.2.4.3

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

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

Step 4.3.3

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

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Divide $3$ by $1$ .

$3$

Step 4.4

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

$μ (x, y) = e^{\int 3 d x}$

Step 5

Evaluate the integral $e^{\int 3 d x}$ .

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Step 5.1

Apply the constant rule.

$μ (x, y) = e^{3 x + C}$

Step 5.2

Simplify.

$μ (x, y) = e^{3 x} + C$

Step 6

Multiply both sides of $(3 x^{2} y + 2 x y + y^{3}) d x + (x^{2} + y^{2}) d y = 0$ by the integration factor $e^{3 x}$ .

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Step 6.1

Multiply $3 x^{2} y + 2 x y + y^{3}$ by $e^{3 x}$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $x^{2} + y^{2}$ by $e^{3 x}$ .

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

Step 6.4

Apply the distributive property.

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

Step 7

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

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

Step 8

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

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Step 8.1

Split the single integral into multiple integrals.

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

Step 8.2

Apply the constant rule.

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Simplify.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

By the Sum Rule, the derivative of $x^{2} e^{3 x} y + \frac{1}{3} e^{3 x} y^{3} + g (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2} e^{3 x} y] + \frac{d}{d x} [\frac{1}{3} e^{3 x} y^{3}] + \frac{d}{d x} [g (x)]$ .

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = e^{3 x}$ .

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

Step 11.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 3 x$ .

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Step 11.3.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

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

Step 11.3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 11.3.3.3

Replace all occurrences of $u_{1}$ with $3 x$ .

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

Step 11.3.4

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

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

Step 11.3.5

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

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

Step 11.3.6

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

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

Step 11.3.7

Multiply $3$ by $1$ .

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

Step 11.3.8

Move $3$ to the left of $e^{3 x}$ .

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

Step 11.4

Evaluate $\frac{d}{d x} [\frac{1}{3} e^{3 x} y^{3}]$ .

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Step 11.4.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

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

Step 11.4.2

Combine $\frac{e^{3 x}}{3}$ and $y^{3}$ .

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

Step 11.4.3

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

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

Step 11.4.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 3 x$ .

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Step 11.4.4.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

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

Step 11.4.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 11.4.4.3

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 11.4.5

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

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

Step 11.4.6

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

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

Step 11.4.7

Multiply $3$ by $1$ .

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

Step 11.4.8

Move $3$ to the left of $e^{3 x}$ .

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

Step 11.4.9

Combine $3$ and $\frac{y^{3}}{3}$ .

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

Step 11.4.10

Combine $\frac{3 y^{3}}{3}$ and $e^{3 x}$ .

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

Step 11.4.11

Cancel the common factor of $3$ .

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Step 11.4.11.1

Cancel the common factor.

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

Step 11.4.11.2

Divide $y^{3} e^{3 x}$ by $1$ .

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

Step 11.5

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

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

Step 11.6

Simplify.

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Step 11.6.1

Apply the distributive property.

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

Step 11.6.2

Remove parentheses.

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

Step 11.6.3

Reorder terms.

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

Step 11.6.4

Reorder factors in $\frac{d g}{d x} + 3 e^{3 x} x^{2} y + 2 e^{3 x} x y + e^{3 x} y^{3}$ .

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

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 $3 x^{2} y e^{3 x}$ from both sides of the equation.

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

Step 12.1.2

Subtract $2 x y e^{3 x}$ from both sides of the equation.

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

Step 12.1.3

Subtract $y^{3} e^{3 x}$ from both sides of the equation.

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

Step 12.1.4

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

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Step 12.1.4.1

Subtract $3 x^{2} y e^{3 x}$ from $3 x^{2} y e^{3 x}$ .

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

Step 12.1.4.2

Add $2 x y e^{3 x} + y^{3} e^{3 x}$ and $0$ .

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

Step 12.1.4.3

Subtract $2 x y e^{3 x}$ from $2 x y e^{3 x}$ .

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

Step 12.1.4.4

Add $y^{3} e^{3 x}$ and $0$ .

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

Step 12.1.4.5

Subtract $y^{3} e^{3 x}$ from $y^{3} e^{3 x}$ .

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

Step 13

Find the antiderivative of $0$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = 0$ .

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

Step 13.2

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

$g (x) = \int 0 d x$

Step 13.3

The integral of $0$ with respect to $x$ is $0$ .

$g (x) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (x) = C$

Step 14

Substitute for $g (x)$ in $f (x, y) = x^{2} e^{3 x} y + \frac{1}{3} e^{3 x} y^{3} + g (x)$ .

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

Step 15

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

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Step 15.1

Simplify each term.

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Step 15.1.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

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

Step 15.1.2

Combine $\frac{e^{3 x}}{3}$ and $y^{3}$ .

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

Step 15.2

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

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