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resuelve la integral por separacion de variables 

To solve the differential equation by separation of variables, we start with the given equation:

\[
\frac{dy}{dx} = \left(\frac{2y + 3}{4x + 5}\right)^2
\]

### Step 1: Separate the variables
We want to isolate all terms involving \( y \) on one side of the equation and all terms involving \( x \) on the other side. We can rearrange the equation as follows:

\[
\frac{dy}{\left(\frac{2y + 3}{4x + 5}\right)^2} = dx
\]

This can be simplified to:

\[
\left(\frac{4x + 5}{2y + 3}\right)^2 dy = dx
\]

### Step 2: Integrate both sides
Now, we integrate both sides of the equation with respect to their respective variables:

\[
\int \left(\frac{4x + 5}{2y + 3}\right)^2 dy = \int dx
\]

### Step 3: Compute the integrals
The integral on the right side is straightforward:

\[
\int dx = x + C_1
\]

For the left side, we integrate with respect to \( y \):

\[
\int \left(\frac{4x + 5}{2y + 3}\right)^2 dy
\]

Since the term \( 4x + 5 \) is constant with respect to \( y \), this can be factored out:

\[
\left(\frac{4x + 5}{2}\right)^2 \int \frac{dy}{(y + \frac{3}{2})^2}
\]

The integral of \( \frac{dy}{(y + \frac{3}{2})^2} \) is:

\[
-\frac{1}{y + \frac{3}{2}}
\]

Thus, the left side becomes:

\[
-\frac{(4x + 5)^2}{2(y + \frac{3}{2})} + C_2
\]

### Step 4: Combine and solve for \( y \)
Putting it all together, we have:

\[
-\frac{(4x + 5)^2}{2(y + \frac{3}{2})} = x + C
\]

where \( C = C_1 - C_2 \) is the combined constant of integration.

Finally, solve for \( y \):

\[
y + \frac{3}{2} = -\frac{(4x + 5)^2}{2(x + C)}
\]

\[
y = -\frac{(4x + 5)^2}{2(x + C)} - \frac{3}{2}
\]

This is the general solution to the differential equation.

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Would you like more details or have any questions about the steps? 

Here are some related questions:

1. How do you find the particular solution if an initial condition is given?
2. Can the separation of variables method be used for all first-order differential equations?
3. What is the geometric interpretation of the solution to this differential equation?
4. How would the solution change if the equation had a different power, say \(\frac{dy}{dx} = \left(\frac{2y + 3}{4x + 5}\right)^n\)?
5. What are some common techniques for integrating more complex expressions after separation of variables?

**Tip:** Always check the domain of the solution, especially when dealing with variables in the denominator. This ensures the solution is valid for all possible values.

Learn how to solve a differential equation using the separation of variables method. Follow our detailed step-by-step guide and understand the integration process involved. Suitable for college-level students and anyone interested in differential equations and calculus.

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