Question

Find CF if (D+D¹-2) (D+2D¹-2)2 = 0

Answer 1

$\text{General partial differential equation is}\\ (c_0D^n+c_1D^{n-1}D^{'}+•••+c_nD^{'n})z=0\\ \text{comparing the given differential equation and the general pde},\\ \text{for }(D+D'-2)(D+2D'-2)^2=0\\ we \,know, when \,aD+bD'+c=0\\ \text{Then, CF is}\space e^{-\frac{c}{a}x}\phi(bx-ay).\\ \text{by comparing}\\ \text{Therefore, CF for this differential equation is }\\e^{2x}(\phi(x-y))+e^x(x\phi_1(2x-y)+\phi_2(2x-y))$