I think we’re just going round in circles without any meaningful progress. In my mind, I’m very clear on where I had reached and where I am stuck. Getting the integrating factor is actually the least of my problems.
I already know how to form the integrating factor — and yes, I agree it’s a bit complex in this case because it involves integrating "ln x". But my real concern is how to move forward from there and actually solve the differential equation, not debating small technical points about the integrating factor.
Now, on the specific issue of whether the constant of integration — the "+ C" — should be included when forming the integrating factor: this is being misunderstood. When we find the integrating factor, we take the exponential of the integral of the function multiplying y. That process does not require including the constant.
Including the constant in the exponent just adds a constant multiple to the whole expression. That constant has no impact on the final solution because it gets absorbed into the constant of integration at the end. That’s why, as a standard method, we intentionally leave it out when calculating the integrating factor. This is not a mistake — it’s how the method is taught and used, including in exams and textbooks.
So if we keep focusing on whether "+ C" is there or not in the integrating factor, we’re missing the real issue. What I need help with is the next part of the process — how to proceed with solving once the integrating factor is applied. That’s where I am genuinely engaging with the math. The rest is just unnecessary back-and-forth over something that doesn’t affect the outcome.