x³\(\frac{d^3y}{dx^3}+3x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+y=\) x + log x---(1)

Let x = e^z ⇒ log x = z

Then x \(\frac{dy}{dx}\) converts into \(\frac{dy}{dx}\) = Dy where D = d/dz

x²\(\frac{d^2y}{dx^2}\) converts into D(D - 1)y

& x³\(\frac{d^3y}{dx^3}\) converts into D(D - 1)(D - 2)y.

Differential equation (1) converts into

D(D - 1)(D - 2)y + 3D(D - 1)y + Dy + y = e^z + z

⇒ (D³ - 3D² + 2D + 3D² - 3D + D + 1)y = e^z + z

⇒ (D³ + 1)y = e^z + z---(2)

It's auxiliarly equation is

m³ + 1 = 0

⇒ (m + 1) (m² - m + 1) = 0

⇒ m = -1, \(\frac{1\pm\sqrt3i}{2}\)

\(\therefore\) C.F. = C₁e^-z + e^z/2(C₂ cos(\(\frac{\sqrt3z}2\)) + C₃ sin(\(\frac{\sqrt3z}2\)))

P.I. = \(\frac1{D^3+1}(e^z+z)\) = \(\frac1{D^3+1}e^z+\frac1{D^3+1}z\) 

= \(\frac{e^z}{1^3+11}+(1+D^3)^{-1}z\) 

= \(\frac{e^z}2+(1-D^3+....)z\) 

= \(\frac{e^z}2+z\) 

\(\therefore\) solution of differential equation (2) is

y = C. F. + P. I.

= C₁e^-z + (e^z)^1/2(C₂ cos(\(\frac{\sqrt3}2\) log x) + C₃ sin(\(\frac{\sqrt3}2\) log x)) + x/2 + log x

which is complete solution of given differential equation.