Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = y (2 x^{2} - x y + 1)$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = 2 x^{2} - x y + 1$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = y (0 + \frac{d}{d y} [- x y] + \frac{d}{d y} [1]) + (2 x^{2} - x y + 1) \frac{d}{d y} [y]$

Step 1.3.3

Add $0$ and $\frac{d}{d y} [- x y]$ .

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

Multiply $- 1$ by $1$ .

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

Step 1.3.7

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = y (- x + 0) + (2 x^{2} - x y + 1) \frac{d}{d y} [y]$

Step 1.3.8

Add $- x$ and $0$ .

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

Step 1.3.9

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

$\frac{\partial M}{\partial y} = y (- x) + (2 x^{2} - x y + 1) \cdot 1$

Step 1.3.10

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

$\frac{\partial M}{\partial y} = y (- x) + 2 x^{2} - x y + 1$

Step 1.4

Subtract $x y$ from $y (- x)$ .

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Step 1.4.1

Move $y$ .

$\frac{\partial M}{\partial y} = 2 x^{2} - x y - x y + 1$

Step 1.4.2

Subtract $x y$ from $- x y$ .

$\frac{\partial M}{\partial y} = 2 x^{2} - 2 x y + 1$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{\partial N}{\partial x} = 1 + \frac{d}{d x} [- y]$

Step 2.4

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

$\frac{\partial N}{\partial x} = 1 + 0$

Step 2.5

Add $1$ and $0$ .

$\frac{\partial N}{\partial x} = 1$

Step 3

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

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Step 3.1

Substitute $2 x^{2} - 2 x y + 1$ for $\frac{\partial M}{\partial y}$ and $1$ for $\frac{\partial N}{\partial x}$ .

$2 x^{2} - 2 x y + 1 = 1$

Step 3.2

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

$2 x^{2} - 2 x y + 1 = 1$ 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 $2 x^{2} - 2 x y + 1$ for $\frac{\partial M}{\partial y}$ .

$\frac{2 x^{2} - 2 x y + 1 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $1$ for $\frac{\partial N}{\partial x}$ .

$\frac{2 x^{2} - 2 x y + 1 - 1}{N}$

Step 4.3

Substitute $x - y$ for $N$ .

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Step 4.3.1

Substitute $x - y$ for $N$ .

$\frac{2 x^{2} - 2 x y + 1 - 1}{x - y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Subtract $1$ from $1$ .

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

Step 4.3.2.2

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

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

Step 4.3.2.3

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

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Step 4.3.2.3.1

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

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

Step 4.3.2.3.2

Factor $2 x$ out of $- 2 x y$ .

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

Step 4.3.2.3.3

Factor $2 x$ out of $2 x (x) + 2 x (- y)$ .

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

Step 4.3.3

Cancel the common factor of $x - y$ .

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Divide $2 x$ by $1$ .

$2 x$

Step 4.4

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

$μ (x, y) = e^{\int 2 x d x}$

Step 5

Evaluate the integral $e^{\int 2 x d x}$ .

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Step 5.1

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

$μ (x, y) = e^{2 \int x d x}$

Step 5.2

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

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

Step 5.3

Simplify the answer.

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Step 5.3.1

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

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

Step 5.3.2

Simplify.

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Step 5.3.2.1

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

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

Step 5.3.2.2

Cancel the common factor of $2$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

$μ (x, y) = e^{1 x^{2}} + C$

Step 5.3.2.3

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

$μ (x, y) = e^{x^{2}} + C$

Step 6

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

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Step 6.1

Multiply $y (2 x^{2} - x y + 1)$ by $e^{x^{2}}$ .

$y (2 x^{2} - x y + 1) e^{x^{2}} d x + (x - y) d y = 0$

Step 6.2

Apply the distributive property.

$(y (2 x^{2}) + y (- x y) + y \cdot 1) e^{x^{2}} d x + (x - y) d y = 0$

Step 6.3

Simplify.

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Step 6.3.1

Rewrite using the commutative property of multiplication.

$(2 y x^{2} + y (- x y) + y \cdot 1) e^{x^{2}} d x + (x - y) d y = 0$

Step 6.3.2

Rewrite using the commutative property of multiplication.

$(2 y x^{2} - y (x y) + y \cdot 1) e^{x^{2}} d x + (x - y) d y = 0$

Step 6.3.3

Multiply $y$ by $1$ .

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

Step 6.4

Multiply $y$ by $y$ by adding the exponents.

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Step 6.4.1

Move $y$ .

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

Step 6.4.2

Multiply $y$ by $y$ .

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

Step 6.5

Apply the distributive property.

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

Step 6.6

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

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

Step 6.7

Apply the distributive property.

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

Step 7

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

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

Step 8

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

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Step 8.1

Split the single integral into multiple integrals.

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

Step 8.2

Apply the constant rule.

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Simplify.

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

Step 8.6

Simplify.

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Step 8.6.1

Multiply $- 1$ by $\frac{1}{2}$ .

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

Step 8.6.2

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

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

Step 8.6.3

Multiply $- 1$ by $\frac{1}{2}$ .

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

Step 8.6.4

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

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

Step 8.6.5

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

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

Step 8.7

Simplify.

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Step 8.7.1

Reorder terms.

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

Step 8.7.2

Remove parentheses.

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

Step 8.7.3

Remove parentheses.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

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 $f (x) = x$ and $g (x) = e^{x^{2}}$ .

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

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 $g (x) = x^{2}$ .

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Step 11.3.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

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 $a$ = $e$ .

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

Step 11.3.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

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

Step 11.3.4

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

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

Step 11.3.5

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

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

Step 11.3.6

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

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

Step 11.3.7

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

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

Step 11.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11.3.9

Add $1$ and $1$ .

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

Step 11.3.10

Move $2$ to the left of $e^{x^{2}}$ .

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

Step 11.3.11

Multiply $e^{x^{2}}$ by $1$ .

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

Step 11.4

Evaluate $\frac{d}{d x} [- \frac{1}{2} y^{2} e^{x^{2}}]$ .

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Step 11.4.1

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

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

Step 11.4.2

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

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

Step 11.4.3

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

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

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 $g (x) = x^{2}$ .

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Step 11.4.4.1

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

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

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 $a$ = $e$ .

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

Step 11.4.4.3

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

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

Step 11.4.5

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

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

Step 11.4.6

Multiply $2$ by $- 1$ .

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

Step 11.4.7

Combine $- 2$ and $\frac{y^{2}}{2}$ .

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

Step 11.4.8

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

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

Step 11.4.9

Combine $\frac{e^{x^{2}} (- 2 y^{2})}{2}$ and $x$ .

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

Step 11.4.10

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

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Step 11.4.10.1

Factor $2$ out of $e^{x^{2}} (- 2 y^{2}) x$ .

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

Step 11.4.10.2

Cancel the common factors.

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Step 11.4.10.2.1

Factor $2$ out of $2$ .

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

Step 11.4.10.2.2

Cancel the common factor.

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

Step 11.4.10.2.3

Rewrite the expression.

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

Step 11.4.10.2.4

Divide $e^{x^{2}} (- y^{2}) x$ by $1$ .

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

Step 11.5

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

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

Step 11.6

Simplify.

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Step 11.6.1

Apply the distributive property.

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

Step 11.6.2

Reorder terms.

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

Step 11.6.3

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

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

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

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

Step 12.1.2

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

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

Step 12.1.3

Add $y^{2} x e^{x^{2}}$ to both sides of the equation.

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

Step 12.1.4

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

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Step 12.1.4.1

Reorder the factors in the terms $2 y x^{2} e^{x^{2}}$ and $- 2 x^{2} y e^{x^{2}}$ .

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

Step 12.1.4.2

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

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

Step 12.1.4.3

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

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

Step 12.1.4.4

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

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

Step 12.1.4.5

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

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

Step 12.1.4.6

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

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

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

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

Step 15

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

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Step 15.1

Simplify each term.

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Step 15.1.1

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

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

Step 15.1.2

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

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

Step 15.2

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

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