2.10 problem 11

Internal problem ID [4892]

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: 11.
ODE order: 2.
ODE degree: 1.

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

Solve \begin {gather*} \boxed {x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (x -2\right ) y=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: 44

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

\[ y \relax (x ) = c_{1} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1+2 x +\frac {3}{2} x^{2}+\frac {2}{3} x^{3}+\frac {5}{24} x^{4}+\frac {1}{20} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]

Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 58

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

\[ y(x)\to c_1 \left (\frac {5 x^3}{24}+\frac {2 x^2}{3}+\frac {3 x}{2}+\frac {1}{x}+2\right )+c_2 \left (\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) \]