20.7 problem 7

Internal problem ID [2471]

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. Additional problems. Section 11.7. page 788
Problem number: 7.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [_Lienard]

Solve \begin {gather*} \boxed {x y^{\prime \prime }+y^{\prime }+y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.016 (sec). Leaf size: 41

Order:=6; 
dsolve(x*diff(y(x),x$2)+diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
 

\[ y \relax (x ) = \left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]

Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 60

AsymptoticDSolveValue[x*y''[x]+y'[x]+x*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (-\frac {3 x^4}{128}+\frac {x^2}{4}+\left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]