18.6 problem (c)

Internal problem ID [2433]

Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number: (c).
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (-2 x^{5}+9 x \right ) y^{\prime }+\left (10 x^{4}+5 x^{2}+25\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.015 (sec). Leaf size: 55

Order:=7; 
dsolve(x^2*diff(y(x),x$2)+(9*x-2*x^5)*diff(y(x),x)+(25+5*x^2+10*x^4)*y(x)=0,y(x),type='series',x=0);
 

\[ y \relax (x ) = c_{1} x^{-4-3 i} \left (1+\left (-\frac {1}{8}-\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}-\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}+\frac {3943 i}{29952}\right ) x^{6}+\mathrm {O}\left (x^{7}\right )\right )+c_{2} x^{-4+3 i} \left (1+\left (-\frac {1}{8}+\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}+\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}-\frac {3943 i}{29952}\right ) x^{6}+\mathrm {O}\left (x^{7}\right )\right ) \]

Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 70

AsymptoticDSolveValue[x^2*y''[x]+(9*x-2*x^5)*y'[x]+(25+5*x^2+10*x^4)*y[x]==0,y[x],{x,0,6}]
 

\[ y(x)\to \left (\frac {1}{832}+\frac {5 i}{832}\right ) c_1 x^{-4+3 i} \left ((86+53 i) x^4+(56+32 i) x^2+(32-160 i)\right )-\left (\frac {5}{832}+\frac {i}{832}\right ) c_2 x^{-4-3 i} \left ((53+86 i) x^4+(32+56 i) x^2-(160-32 i)\right ) \]