Find the general solution of the given differential equation. x²y' + xy = 9

Solution:

Given, the differential equation is x²y' + xy = 9

We have to find the general solution of the given differential equation.

The equation can be written as x²(dy/dx) + xy = 9

Dividing by x² on both sides,

dy/dx + y/x = 9/x²

This represents the form dy/dx + P(x)y = Q(x), which is a linear differential equation

Here, P(x) = 1/x and Q(x) = 9/x².

Integrating factor (I. F.) = e ^∫P(x). dx

I.F. = e ^∫1/x. dx

= e^logex

= x

The solution of Linear differential equation is given by

(I.F.) × y = ∫ (I.F.) Q(x) dx

xy = ∫ x. (9/x²) dx

xy = 9∫(x/x²) dx

xy = 9 ∫ (1/x) . dx

xy = 9 log_ex + C

Therefore, the required solution is xy = 9 log_ex + C

---

Find the general solution of the given differential equation. x²y' + xy = 9

Summary:

The general solution of the differential equation. x²y' + xy = 9 is xy = 9 log_ex + C.