Internal problem ID [1821]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.3, The method of Frobenius. Equal roots, and roots differering by an integer. Page
223
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {t^{2} y^{\prime \prime }-t \left (t +1\right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 63
Order:=6; dsolve(t^2*diff(y(t),t$2)-t*(1+t)*diff(y(t),t)+y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = t \left (\left (c_{2} \ln \relax (t )+c_{1}\right ) \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (-t -\frac {3}{4} t^{2}-\frac {11}{36} t^{3}-\frac {25}{288} t^{4}-\frac {137}{7200} t^{5}+\mathrm {O}\left (t^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 112
AsymptoticDSolveValue[t^2*y''[t]-t*(1+t)*y'[t]+y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 t \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right )+c_2 \left (t \left (-\frac {137 t^5}{7200}-\frac {25 t^4}{288}-\frac {11 t^3}{36}-\frac {3 t^2}{4}-t\right )+t \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right ) \log (t)\right ) \]