In Green's method to solve non homogeneous IVP : $$a(x)y''(x)+b(x)y'+c(x)y=f(x),\\y(0)=y_o,y'(0)=v_o$$ We divide the problem to 2 sub-problems :

We assume that the solution of the homogeneous problem satisfies the original initial conditions: $$a(x)y_h''(x)+b(x)y_h'+c(x)y_h=0,\\y_h(0)=y_o,y_h'(0)=v_o$$ We then assume that the particular solution satisfies the problem $$a(x)y_p''(x)+b(x)y_p'+c(x)y_p=f(x),\\y_p(0)=0,y_p'(0)=0$$ using green's method : yp is $$y_p(x)=\int_{0}^{x}{G(x,t)f(t)dt}$$ $$G(x,t)=\frac{y_1(t)y_2(x)-y_2(t)y_1(x)}{a(t)W(t)}$$ hence $$y(x)=y_h(x)+y_p(x)$$ Note: the proof and formulas are from this link : http://people.uncw.edu/hermanr/pde1/PDEbook/Green.pdf

My question is:

why do we call the solution of the second subproblem ( the non homogeneous equation with homogeneous initial conditions) a particular solution although this second problem if solved using undetermined coefficient for example we will get homogeneous solution and a particular solution not only a particular solution!

[ I want to check whether I understand right or not .. it is just called a particular solution although it is not , right ?! so why it is called in this way ?!]

for example the problem $$y''-3y'+2y=20e^{-2x}$$ for the first subproblem we get $$y_h(x)=-6(e^{x}-e^{2x})$$ for the second subproblem and using green's function $$y_p(x)=\frac{5}{3}e^{-2x}+5e^{2x}-\frac{20}{3}e^{x}$$ so the final solution is the summation of yh and yp $$y(x)=y_h(x)+y_p(x)=\frac{-38}{3}e^{x}+11e^{2x}+\frac{5}{3}e^{-2x}$$ However ,if we solve the second subproblem using undetermined coefficients , we get first an homog solution then a particular solution and we add them $$y(x)=c_1e^x+c_2e^{2x}+\frac{5}{3}e^{-2x}$$ Applying the zero initial conditions: $$y(x)=\frac{-20}{3}e^x+5e^{2x}+\frac{5}{3}e^{-2x}$$ So the second subproblem itself has an homogeneous and a particular solution not only a particular solution .


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