Heres another example (2x-1)dx+((3y+7)dy=0

To solve the differential equation \((2x - 1) \, dx + (3y + 7) \, dy = 0\), we'll follow these steps:

### Step 1: Rewrite the Equation
Rewriting the given differential equation:
\[
(2x - 1) \, dx + (3y + 7) \, dy = 0.
\]

### Step 2: Separate the Variables
To separate the variables, we can rearrange the equation as:
\[
\frac{dx}{3y + 7} = -\frac{dy}{2x - 1}.
\]

### Step 3: Integrate Both Sides
Now, integrate both sides:

For the left side (in terms of \(x\)):
\[
\int \frac{1}{2x - 1} \, dx.
\]

For the right side (in terms of \(y\)):
\[
\int \frac{1}{3y + 7} \, dy.
\]

**Integrating:**
\[
\int \frac{1}{2x - 1} \, dx = \frac{1}{2} \ln |2x - 1| + C_1,
\]
\[
\int \frac{1}{3y + 7} \, dy = \frac{1}{3} \ln |3y + 7| + C_2.
\]

### Step 4: Combine the Results
Setting the integrals equal to each other:
\[
\frac{1}{2} \ln |2x - 1| + C_1 = -\left(\frac{1}{3} \ln |3y + 7| + C_2\right).
\]

Simplifying, we get:
\[
\frac{1}{2} \ln |2x - 1| + \frac{1}{3} \ln |3y + 7| = C,
\]
where \(C = C_2 - C_1\).

Alternatively, we can express the solution implicitly by exponentiating both sides:
\[
|2x - 1|^{1/2} \cdot |3y + 7|^{1/3} = e^C.
\]

Simplifying further:
\[
|2x - 1|^{1/2} \cdot |3y + 7|^{1/3} = K,
\]
where \(K = e^C\) is a constant.

Thus, the general solution is:
\[
|2x - 1|^{1/2} \cdot |3y + 7|^{1/3} = K.
\]

Would you like more details or have any specific questions?

### Related Questions:
1. How do you solve a differential equation if variables cannot be separated easily?
2. What are the methods to solve non-separable differential equations?
3. How to verify the solution of a differential equation?
4. Can we solve this differential equation using an integrating factor?
5. How do constants affect the general solution of differential equations?

**Tip:** Always check if a differential equation can be separated before trying more complex methods, as separable equations are generally simpler to solve!

Learn how to solve the differential equation (2x - 1)dx + (3y + 7)dy = 0 with a step-by-step solution. This problem covers differential equations, separable equations, and integration techniques. Suitable for advanced high school students and beyond.

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