Find the general solution of the given differential equation. x²y' + xy = 4.

Solution:

Given, x²y' + xy = 4

⇒ x² .dy/dx + x . y = 4

Divide both the sides by x²

dy/dx + y/x = 4/x²

It is of the form dy/dx + P(x)y = Q(x), which is a linear differential equation where, P(x) = 1/x and Q(x) = 4/x².

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

I.F. = e ^∫1/x. dx

= e^log_ex

= x

Solution of Linear differential equation is 

(I.F.) × y = ∫ (I.F.) Q(x) dx

x y = ∫ x. (4/x²) dx

xy = 4 ∫ (1/x) . dx

xy = 4 ^log_ex + C

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Find the general solution of the given differential equation. x²y' + xy = 4.

Summary:

The general solution of the given differential equation. x²y' + xy = 4 is xy = 4 ^log_ex + C.