Recall the following theorem:
Suppose p and q are continuous on an open interval (a,b). Let x_(0) be any point in
(a,b), and let k_(0) and k_(1) be arbitrary real numbers. Then the initial value problem
y^('')+p(x)y^(')+q(x)y=0,y(x_(0))=k_(0),y^(')(x_(0))=k_(1)
has a unique solution on (a,b).
Assuming that p and q are continuous on an open interval (a,b) and x_(0)in(a,b), give a detailed
proof that the only solution of the initial value problem
y^('')+p(x)y^(')+q(x)y=0,y(x_(0))=0,y^(')(x_(0))=0
on (a,b) is the trivial solution y-=0. Your work should be legible, and all your logic should be
clear and justified.
Recall the following theorem:
Suppose p and q are continuous on an open interval (a, b). Let o be any point in a, b), and let ko and k1 be arbitrary real numbers. Then the initial value problem y"+p(x)y+q(x)y=0, y(xo)=ko, y(xo)=k1 has a unique solution on (a, b).
Assuming that p and q are continuous on an open interval (a, b) and o E (a, b), give a detailed proof that the only solution of the initial value problem y"+p(x)y'+q(x)y=0, y(xo)=0, y'(xo)=0 on (a, b) is the trivial solution y = 0. Your work should be legible, and all your logic should be clear and justified