For each of the following IVP determine the first four nonze

For each of the following IVP. determine the first four non-zero terms in. as well as the radius of convergence of the power series of the solution y = y(x) c^zy\" + xy = 0, y(0) = 1, y\'(0) = 1 y\" - xy\' - y = 2 sin x, y(0) = 1, y\'(0) = 0

Solution

e^x y\'\'+ xy=0, ....................(1)

y(0)=1, y\'(0) =1

So we assume y(x) = 1+x+ a[2]x² +a[3]x³ +.................................(2)

Substituting in (1) and equating coefficients (after expanding e^x in power series)

we obtain

                      a[2]=a[4]=0

                   a[3] = -1/6, a[5] =-1/40

So the first four (n0nvanishing) terms of the powerseries solution are 1+x-1/6x³ -1/40 x⁵

Radius of convergence 1

(ii) As y(o) =1, y\'(0) =0 ,

.        we take y(x) = 1+ a[2]x² +a[3]x³ +a[4]x⁴ +.......

Equating the coefficients after substituting in the equation (and using the powerseries for sin x), we get

2a[2]-1=0, so a[2]=1/2

6a[3] =2 so a[3]=1/3

So the powerseries solution starts with 1+x+x²/2+x³ /3

As the recurrence relation coeffients have (of the order of n!) in the denominator (due to sin x)) , the radius of convergence is infinity.