Find the general solution of the given differential equation. x²y' + xy = 8. y(x) = give the largest interval over which the general solution is defined.

Solution:

Given, x²y' + xy = 8

⇒ x² .dy/dx + x . y = 8

Divide both the sides by x²

dy / dx + y / x = 8 / 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. (8 / x²) dx

xy = 8 ∫ (1 / x) . dx

xy = 8 log_ex + C

This general solution is defined as ∀ x ϵ R⁺ because if x = 0 or x = -ve, log_ex does not exist.

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Find the general solution of the given differential equation. x²y' + xy = 8. y(x) = give the largest interval over which the general solution is defined.

Summary:

The general solution of the given differential equation. x²y' + xy = 8 is xy = 8 log_ex + C.