Question

Determine whether or not each of the equations is exact. If it is exact, find the solution. $$ \left(y e^{x y} \cos 2 x-2 e^{x y} \sin 2 x+2 x\right) d x+\left(x e^{x y} \cos 2 x-3\right) d y=0 $$

Short answer

#Answer# The general solution of the above exact differential equation is: $$ e^{xy} (y \cos 2x - 2 \sin 2x) + x^2 + C(y) = e^{xy} (x \cos 2x) - 3y + D(x) $$

Step-by-step solution

Check if the given equation is exact

First, compute the partial derivatives of M with respect to y and N with respect to x: $$ \frac{\partial M}{\partial y} = e^{xy} \cos 2x + xye^{xy} \cos 2x - 2xe^{xy} \sin 2x \\ \frac{\partial N}{\partial x} = e^{xy} \cos 2x + xye^{xy} \cos 2x - 2 ye^{xy} \sin 2x + 2x $$ Since: $$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} $$ The given differential equation is exact.

Integrate M in terms of x

Now, integrate M with respect to x to find the first part of the solution: $$ \int M(x, y) dx = \int (y e^{xy} \cos 2x-2 e^{xy} \sin 2x+2 x) dx $$ Let this integral be equal to F(x, y): $$ F(x, y) = e^{xy} (y \cos 2x - 2 \sin 2x) + x^2 + C(y) $$

Integrate N in terms of y

Now, we need to integrate N with respect to y: $$ \int N(x, y) dy = \int (x e^{xy} \cos 2x - 3) dy $$ Let this integral be equal to G(x, y): $$ G(x, y) = e^{xy} (x \cos 2x) - 3y + D(x) $$

Determine the general solution

From the integrals of M and N, we have the following general solutions: $$ F(x, y) = e^{xy} (y \cos 2x - 2 \sin 2x) + x^2 + C(y) \\ G(x, y) = e^{xy} (x \cos 2x) - 3y + D(x) $$ The general solution of the exact differential equation is obtained by comparing these two expressions: $$ F(x, y) = G(x, y) $$ Which gives the following general solution: $$ e^{xy} (y \cos 2x - 2 \sin 2x) + x^2 + C(y) = e^{xy} (x \cos 2x) - 3y + D(x) $$

Key concepts

Partial Derivatives

Understanding partial derivatives is essential when dealing with exact differential equations. Let's imagine a function that depends on two variables, like temperature might depend on both time and position. Whenever we want to see how the function changes in just one of those variables, while keeping the other constant, we're looking at its partial derivative.

For the equation at hand, we compute the partial derivatives of two bordering components: one with respect to y (written as \( \frac{\partial M}{\partial y} \)) and the other with respect to x (expressed as \( \frac{\partial N}{\partial x} \)). By comparing them, we can determine if the equation is exact—that is, if these partial derivatives are equal. In our case, they are, confirming the exactness of the equation.

Integration of Differential Equations

Now, consider the integration of differential equations. Integration is, perhaps, most easily thought of as the reverse process of differentiation: if differentiation gives us the rate at which something changes, integration tells us the total change over an interval.

In the provided exercise, we integrate the components of the equation (labeled M and N) separately with respect to x and y to work towards our solution.

Integrating M with respect to x

We obtain function \( F(x, y) \) by integrating M over x, which, metaphorically, gives us a 'partial picture' of the total change.

Integrating N with respect to y

Similarly, integrating N over y yields another function \( G(x, y) \), helping us fill in another part of the solution's landscape.

General Solution of Differential Equations

Finally, we reach the general solution of differential equations. After integrating the separate pieces of our exact differential equation, we want to combine these to find a comprehensive expression that can tell us about the behavior of the system for various x and y. In simple terms, it's like assembling a puzzle with the pieces we've obtained from integration.

By equating \( F(x, y) \) and \( G(x, y) \), which we found by separately integrating M and N, we arrive at the general solution of the equation. This solution is not just a single answer but a family of curves or surfaces in the space defined by x and y that describe the system's behavior under a whole range of conditions. The idea here is like having a map of potential routes rather than a set direction—each specific solution being a path tailored to certain initial or boundary conditions.