Calculus Examples

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

Step 1

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

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Step 1.1

Rewrite.

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

Step 2

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

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Step 2.1

Differentiate $M$ with respect to $y$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $2$ by $1$ .

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

Step 2.4

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

$\frac{\partial M}{\partial y} = 1 + 2 x^{2} + 0$

Step 2.5

Simplify.

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Step 2.5.1

Add $1 + 2 x^{2}$ and $0$ .

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

Step 2.5.2

Reorder terms.

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

Step 3

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

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Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

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

Step 5.2

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

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

Step 5.3

Substitute $x$ for $N$ .

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Step 5.3.1

Substitute $x$ for $N$ .

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

Step 5.3.2

Simplify the numerator.

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Step 5.3.2.1

Subtract $1$ from $1$ .

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

Step 5.3.2.2

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

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

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

Step 5.3.3.2.2

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

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

Step 5.3.3.2.3

Cancel the common factor.

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

Step 5.3.3.2.4

Rewrite the expression.

$\frac{2 x}{1}$

Step 5.3.3.2.5

Divide $2 x$ by $1$ .

$2 x$

Step 5.4

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

$μ (x, y) = e^{\int 2 x d x}$

Step 6

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

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Step 6.1

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

$μ (x, y) = e^{2 \int x d x}$

Step 6.2

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 6.3

Simplify the answer.

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Step 6.3.1

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 6.3.2

Simplify.

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Step 6.3.2.1

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

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

Step 6.3.2.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1

Cancel the common factor.

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

Step 6.3.2.2.2

Rewrite the expression.

$μ (x, y) = e^{1 x^{2}} + C$

Step 6.3.2.3

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

$μ (x, y) = e^{x^{2}} + C$

Step 7

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

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Step 7.1

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

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

Step 7.2

Apply the distributive property.

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

Step 7.3

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

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

Step 8

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

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

Step 9

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

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Step 9.1

Apply the constant rule.

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

Differentiate $f$ with respect to $x$ .

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

Step 12.2

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

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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 12.3.3.1

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

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

Step 12.3.3.2

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

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

Step 12.3.3.3

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

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

Step 12.3.4

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

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

Step 12.3.5

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

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

Step 12.3.6

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

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

Step 12.3.7

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

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

Step 12.3.8

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

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

Step 12.3.9

Add $1$ and $1$ .

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

Step 12.3.10

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

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

Step 12.3.11

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

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

Step 12.4

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

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

Step 12.5

Simplify.

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Step 12.5.1

Apply the distributive property.

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

Step 12.5.2

Reorder terms.

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

Step 12.5.3

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

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

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

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

Step 13.1.2

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

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

Step 13.1.3

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

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Step 13.1.3.1

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

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

Step 13.1.3.2

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

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

Step 13.1.3.3

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

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

Step 13.1.3.4

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

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

Step 13.1.3.5

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

Let $u_{2} = e^{x^{2}}$ . Then $d u_{2} = 2 x e^{x^{2}} d x$ , so $\frac{1}{2} d u_{2} = x e^{x^{2}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 14.4.1

Let $u_{2} = e^{x^{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 14.4.1.1

Differentiate $e^{x^{2}}$ .

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

Step 14.4.1.2

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 14.4.1.2.1

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

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x^{2}]$

Step 14.4.1.2.2

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

$e^{u_{1}} \frac{d}{d x} [x^{2}]$

Step 14.4.1.2.3

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

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

Step 14.4.1.3

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

$e^{x^{2}} (2 x)$

Step 14.4.1.4

Simplify.

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Step 14.4.1.4.1

Reorder the factors of $e^{x^{2}} (2 x)$ .

$2 e^{x^{2}} x$

Step 14.4.1.4.2

Reorder factors in $2 e^{x^{2}} x$ .

$2 x e^{x^{2}}$

Step 14.4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$g (x) = - 2 \int \frac{1}{2} d u_{2}$

Step 14.5

Apply the constant rule.

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

Step 14.6

Simplify the answer.

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Step 14.6.1

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

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

Step 14.6.2

Simplify.

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Step 14.6.2.1

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

$g (x) = \frac{- 2}{2} u_{2} + C$

Step 14.6.2.2

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

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Step 14.6.2.2.1

Factor $2$ out of $- 2$ .

$g (x) = \frac{2 \cdot - 1}{2} u_{2} + C$

Step 14.6.2.2.2

Cancel the common factors.

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Step 14.6.2.2.2.1

Factor $2$ out of $2$ .

$g (x) = \frac{2 \cdot - 1}{2 (1)} u_{2} + C$

Step 14.6.2.2.2.2

Cancel the common factor.

$g (x) = \frac{2 \cdot - 1}{2 \cdot 1} u_{2} + C$

Step 14.6.2.2.2.3

Rewrite the expression.

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

Step 14.6.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 14.6.3

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

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

Step 15

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

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

Step 16

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

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