Q - 8EExpert-verified
Question: In Problems 1–10, determine all the singular points of the given differential equation.
There is no singularity point that exists in this differential equation for both P(X) and Q(X).
A point X0 is called an ordinary point of equation y"+p(x)y'+q(x)y=0 if both p and q are analytic at X0 . If X0 is not an ordinary point, it is called a singular point of the equation.
The given differential equation is
Dividing the above equation by ex we get,
On comparing the above equation with y"+p(x)y'+q(x)y=0 ,we find that,
Hence, Px) and Q(x) are analytic except, perhaps, when their denominators are zero.
For Px) this occurs at no point. In this case both Px) and Q(x) are analytic at all points.
Therefore, there is no singularity point that exists in this differential equation for both Px) and Q(x) .
To derive the general solutions given by equations (17)- (20) for the non-homogeneous equation (16), complete the following steps.
(a) Substitute and the Maclaurin series into equation (16) to obtain
(b) Equate the coefficients of like powers on both sides of the equation in part (a) and thereby deduce the equations
(c) Show that the relations in part (b) yield the general solution to (16) given in equations (17)-(20).
Show that fn(0)=0 for n=0,1,2.... and hence that the Maclaurin series for f(x) is 0+0+0+.... , which converges for all x but is equal to f(x) only when x=0 . This is an example of a function possessing derivatives of all orders (at x0 =0 ), whose Taylor series converges, but the Taylor series (about x0 =0) does not converge to the original function! Consequently, this function is not analytic at x=0.
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