2.2 problem 3

Internal problem ID [4884]

Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number: 3.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [_Lienard]

Solve \begin {gather*} \boxed {x y^{\prime \prime }+2 y^{\prime }+y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.078 (sec). Leaf size: 32

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

\[ y \relax (x ) = c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]

Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 42

AsymptoticDSolveValue[x*y''[x]+2*y'[x]+x*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{24}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^4}{120}-\frac {x^2}{6}+1\right ) \]