# About Green function on ODE IVP, BVP

I am sophomore student learning ODE.

While learning ODE, suddenly met Green function in IVP, BVP.

My 1st question is why it is introduced in IVP, BVP, such as: (Due to reduction of order & initial, boundary conditions??)

G(x,t)= $$\begin{cases} { y_1(t) y_2(x)\over W(t)} & \mbox{if } a\leq t\leq x \mbox{} \\ { y_1(x) y_2(t)\over W(t)}& \mbox{if } x\leq t\leq b\mbox{ } \end{cases}$$

Second, How can I understand that G(x,t) is dependent only on y1 (x), y2(x), but independent on f(x)? ( From the Initial or boundary values, get its Wronskian? ) ( From my book, A first course in diffrential equations 11th editions, Dennis G. Zill Chapter 4.8)

1st, 2nd questions might be related. Even though the answer is one, I wanted to ask whether it's the strongest, easiest way to use on IVP-BVP, compared to the others.

I'd really appreciate your sincere answer. My knowledge is just average sophomore early math major student.

Thank you so much for sparing your time.

• Hi and welcome to Math.SE. It would be preferable to use MathJax for mathematical expressions. You can get started here, and a more complete reference can be found here. May 13 at 9:42
• @user3733558 Thank you so much May 13 at 10:51

We have a linear differential operator $$L$$, $$L[y]=y''+py'+qy$$, and want to solve $$L[y]=f$$ with homogeneous boundary conditions. If all the right sides $$f$$ that are "admissible" in some sense (for instance piecewise linear over a fixed subdivision) can be represented as a linear combinations of "atom" or kernel functions $$f(x)=\sum_{k=1}^Nc_kg_k(x)$$ then one would only have to solve $$L[y_k]=g_k$$ to construct any solution as $$y(x)=\sum_{k=1}^N c_ky_k(x)$$. So one can solve a large number of problems with the solutions of a finite number of BVPs.
If one carries the atomic decomposition construction to its extreme, then one ends up with the "sifting" property $$f=\delta*f=\int_a^bf(s)\delta_s\,ds$$. This reduces the solution of $$L[y]=f$$ for a general right side to the solution of $$L[y_s]=δ_s$$ for any $$s\in[a,b]$$. Then the general solution can be reconstructed by "summation" over $$s$$, $$y(x)=\int_a^b f(s)y_s(x)\,ds$$. Making $$s$$ into a function argument, this gives the Green function $$G(x,s)=y_s(x)$$.
Because of the properties of the Dirac delta distribution, one has $$L[y_s](x)=0$$ for $$x\ne s$$ and $$y_s$$ satisfies the homogeneous boundary conditions. Thus the separation into parts over $$[a,s]$$ and $$[s,b]$$ that are homogeneous solutions, at first independent. The continuity of the solution at $$x=s$$ then requires that $$y_s(x)=C(s)·y_1(\min(s,x))·y_2(\max(s,x))$$ where $$y_1$$ satisfies the left and $$y_2$$ the right boundary condition. Note that only one factor is "active", the other two are constants. The value of $$C(s)$$ follows from the condition that the first derivative $$y_s'$$ needs to have a unit jump at $$x=s$$, $$y_s'(x)=C(s)·[y_1'(\min(s,x))·u(s-x)·y_2(\max(s,x))+y_1(\min(s,x))·y_2'(\max(s,x))·u(x-s)],\\ 1=y_s'(s+0)-y_s'(s-0)=C(s)·[y_1(s)·y_2'(s)-y_1'(s)·y_2(s)]=C(s)·W[y_1,y_2](s).$$