Internal problem ID [1804]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 t^{2} y^{\prime \prime }+\left (t^{2}-t \right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 45
Order:=6; dsolve(2*t^2*diff(y(t),t$2)+(t^2-t)*diff(y(t),t)+y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} \sqrt {t}\, \left (1-\frac {1}{2} t +\frac {1}{8} t^{2}-\frac {1}{48} t^{3}+\frac {1}{384} t^{4}-\frac {1}{3840} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} t \left (1-\frac {1}{3} t +\frac {1}{15} t^{2}-\frac {1}{105} t^{3}+\frac {1}{945} t^{4}-\frac {1}{10395} t^{5}+\mathrm {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 86
AsymptoticDSolveValue[2*t^2*y''[t]+(t^2-t)*y'[t]+y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 t \left (-\frac {t^5}{10395}+\frac {t^4}{945}-\frac {t^3}{105}+\frac {t^2}{15}-\frac {t}{3}+1\right )+c_2 \sqrt {t} \left (-\frac {t^5}{3840}+\frac {t^4}{384}-\frac {t^3}{48}+\frac {t^2}{8}-\frac {t}{2}+1\right ) \]