To determine the general solution (PU) for the differential equation:

$(x + 2y + 1)dx - (2x - 3)dy = 0$

Step 1: Verify if the Equation is Exact

The differential equation is of the form:

$M(x, y) \, dx + N(x, y) \, dy = 0$

where:

• $M(x, y) = x + 2y + 1$
• $N(x, y) = -(2x - 3) = -2x + 3$

To check if this equation is exact, we need to compute the partial derivatives:

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

$\frac{\partial N}{\partial x} = \frac{\partial (-2x + 3)}{\partial x} = -2$

Since $\frac{\partial M}{\partial y} = 2$ and $\frac{\partial N}{\partial x} = -2$, and these are not equal, the equation is not exact.

Step 2: Finding the Integrating Factor

Given the non-exact nature of the equation, we attempt to find an integrating factor. Since the equation depends on both $x$ and $y$, we should look for a suitable integrating factor that could depend on $x$ or $y$.

One potential strategy is to find an integrating factor $\mu(x)$ or $\mu(y)$ such that multiplying the entire equation by $\mu$ makes it exact. However, without a clear dependency that simplifies to a known form, it can be difficult to find such a factor using simple methods.

Step 3: Assume and Solve (Direct Method)

Let's try to find a solution by assuming that there exists a potential function $U(x, y)$ such that:

$\frac{\partial U}{\partial x} = M(x, y) = x + 2y + 1$ $\frac{\partial U}{\partial y} = N(x, y) = -2x + 3$

Now, integrate $M(x, y)$ with respect to $x$:

$U(x, y) = \int (x + 2y + 1) \, dx = \frac{x^2}{2} + (2y + 1)x + g(y)$

where $g(y)$ is an arbitrary function of $y$ that appears since the integration is with respect to $x$.

Now, differentiate this result with respect to $y$ and equate it to $N(x, y)$:

$\frac{\partial U}{\partial y} = 2x + g'(y)$

Set this equal to $N(x, y) = -2x + 3$:

$2x + g'(y) = -2x + 3$

Solving for $g'(y)$, we get:

$g'(y) = 3 - 4x$

Since $g'(y)$ should be a function of $y$ only, this implies that we must have made an error in our assumption or method. This suggests that either the integrating factor approach or another method must be employed to find a solution.

Given this complexity, it's clear that this differential equation may require a more advanced method to solve or to identify a suitable integrating factor.

Conclusion

The differential equation provided is non-exact, and finding a general solution may require an integrating factor or a numerical approach that goes beyond elementary methods. Would you like to explore this further, perhaps with specific assumptions or numerical methods?

Follow-Up Questions:

1. What specific methods can be used to find an integrating factor for this type of non-exact differential equation?
2. How does the presence of a non-exact equation affect the process of solving it?
3. Are there other examples of non-exact differential equations that can be solved using elementary techniques?
4. How can numerical methods assist in solving non-exact differential equations?
5. What role does the integrating factor play in converting non-exact equations into exact ones?

Tip:

When dealing with non-exact equations, always check simple potential integrating factors like functions of $x$ or $y$ alone before considering more complex dependencies.