Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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 $g (y) = x^{2} - y + x$ .

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

Step 1.4

Differentiate.

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- y$ with respect to $y$ is $- \frac{d}{d y} [y]$ .

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

Step 1.4.5

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

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

Step 1.4.6

Multiply $- 1$ by $1$ .

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

Step 1.4.7

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

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

Step 1.4.8

Simplify the expression.

Tap for more steps...

Step 1.4.8.1

Add $- 1$ and $0$ .

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

Step 1.4.8.2

Move $- 1$ to the left of $y$ .

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

Step 1.4.8.3

Rewrite $- 1 y$ as $- y$ .

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

Step 1.4.9

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

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

Step 1.4.10

Simplify by adding terms.

Tap for more steps...

Step 1.4.10.1

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

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

Step 1.4.10.2

Subtract $y$ from $- y$ .

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

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Multiply $- 2$ by $2$ .

$\frac{\partial M}{\partial y} = 2 e^{2 x} x^{2} - 4 e^{2 x} y + 2 e^{2 x} x$

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Add $2 x$ and $0$ .

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

Step 2.4

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 $g (x) = 2 x$ .

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2.5

Differentiate.

Tap for more steps...

Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

Simplify the expression.

Tap for more steps...

Step 2.5.3.1

Multiply $2$ by $1$ .

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

Step 2.5.3.2

Move $2$ to the left of $(x^{2} - 2 y) e^{2 x}$ .

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

Step 2.6

Simplify.

Tap for more steps...

Step 2.6.1

Apply the distributive property.

$\frac{\partial N}{\partial x} = e^{2 x} (2 x) + (2 x^{2} + 2 (- 2 y)) e^{2 x}$

Step 2.6.2

Apply the distributive property.

$\frac{\partial N}{\partial x} = e^{2 x} (2 x) + 2 x^{2} e^{2 x} + 2 (- 2 y) e^{2 x}$

Step 2.6.3

Multiply $- 2$ by $2$ .

$\frac{\partial N}{\partial x} = e^{2 x} (2 x) + 2 x^{2} e^{2 x} - 4 y e^{2 x}$

Step 2.6.4

Reorder terms.

$\frac{\partial N}{\partial x} = 2 e^{2 x} x + 2 e^{2 x} x^{2} - 4 e^{2 x} y$

Step 2.6.5

Reorder factors in $2 e^{2 x} x + 2 e^{2 x} x^{2} - 4 e^{2 x} y$ .

$\frac{\partial N}{\partial x} = 2 x e^{2 x} + 2 x^{2} e^{2 x} - 4 y e^{2 x}$

Step 3

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

Tap for more steps...

Step 3.1

Substitute $2 e^{2 x} x^{2} - 4 e^{2 x} y + 2 e^{2 x} x$ for $\frac{\partial M}{\partial y}$ and $2 x e^{2 x} + 2 x^{2} e^{2 x} - 4 y e^{2 x}$ for $\frac{\partial N}{\partial x}$ .

$2 e^{2 x} x^{2} - 4 e^{2 x} y + 2 e^{2 x} x = 2 x e^{2 x} + 2 x^{2} e^{2 x} - 4 y e^{2 x}$

Step 3.2

Since the two sides have been shown to be equivalent, the equation is an identity.

$2 e^{2 x} x^{2} - 4 e^{2 x} y + 2 e^{2 x} x = 2 x e^{2 x} + 2 x^{2} e^{2 x} - 4 y e^{2 x}$ is an identity.

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

Split the single integral into multiple integrals.

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

Step 5.3

Apply the constant rule.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Simplify.

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

Step 6

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

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

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

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 $g (x) = 2 x$ .

Tap for more steps...

Step 8.3.6.1

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

$e^{2 x} (y (2 x) + 0) + (x^{2} y - y^{2}) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]) + \frac{d}{d x} [g (x)] = 2 y e^{2 x} (x^{2} - y + x)$

Step 8.3.6.2

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

$e^{2 x} (y (2 x) + 0) + (x^{2} y - y^{2}) (e^{u} \frac{d}{d x} [2 x]) + \frac{d}{d x} [g (x)] = 2 y e^{2 x} (x^{2} - y + x)$

Step 8.3.6.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 8.3.7

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

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

Step 8.3.8

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

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

Step 8.3.9

Move $2$ to the left of $y$ .

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

Step 8.3.10

Add $2 y x$ and $0$ .

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

Step 8.3.11

Multiply $2$ by $1$ .

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

Step 8.3.12

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

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

Step 8.3.13

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

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

Step 8.4

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

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

Step 8.5

Simplify.

Tap for more steps...

Step 8.5.1

Apply the distributive property.

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

Step 8.5.2

Apply the distributive property.

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

Step 8.5.3

Multiply $- 1$ by $2$ .

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

Step 8.5.4

Reorder terms.

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

Step 9

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

Tap for more steps...

Step 9.1

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

Tap for more steps...

Step 9.1.1

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

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

Step 9.1.2

Simplify $2 y e^{2 x} (x^{2} - y + x)$ .

Tap for more steps...

Step 9.1.2.1

Apply the distributive property.

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

Step 9.1.2.2

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

Tap for more steps...

Step 9.1.2.2.1

Move $y$ .

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

Step 9.1.2.2.2

Multiply $y$ by $y$ .

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

Step 9.1.2.3

Multiply $- 1$ by $2$ .

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

Step 9.1.2.4

Reorder factors in $2 y e^{2 x} x^{2} - 2 y^{2} e^{2 x} + 2 y e^{2 x} x$ .

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

Step 9.1.3

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

Tap for more steps...

Step 9.1.3.1

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

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

Step 9.1.3.2

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

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

Step 9.1.3.3

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

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

Step 9.1.3.4

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

Tap for more steps...

Step 9.1.3.4.1

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

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

Step 9.1.3.4.2

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

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

Step 9.1.3.4.3

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

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

Step 9.1.3.4.4

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

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

Step 9.1.3.4.5

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

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

Step 9.1.3.4.6

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

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

Step 9.1.3.4.7

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

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

Step 10

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

Tap for more steps...

Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

$g (x) = 0 + C$

Step 10.4

Add $0$ and $C$ .

$g (x) = C$

Step 11

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

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

Step 12

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

Tap for more steps...

Step 12.1

Simplify each term.

Tap for more steps...

Step 12.1.1

Apply the distributive property.

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

Step 12.1.2

Rewrite using the commutative property of multiplication.

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

Step 12.2

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

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