Internal problem ID [1801]
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: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 t y^{\prime \prime }+\left (t +1\right ) y^{\prime }-2 y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 38
Order:=6; dsolve(2*t*diff(y(t),t$2)+(1+t)*diff(y(t),t)-2*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} \sqrt {t}\, \left (1+\frac {1}{2} t +\frac {1}{40} t^{2}-\frac {1}{1680} t^{3}+\frac {1}{40320} t^{4}-\frac {1}{887040} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} \left (1+2 t +\frac {1}{3} t^{2}+\mathrm {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 62
AsymptoticDSolveValue[2*t*y''[t]+(1+t)*y'[t]-2*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 \left (\frac {t^2}{3}+2 t+1\right )+c_1 \sqrt {t} \left (-\frac {t^5}{887040}+\frac {t^4}{40320}-\frac {t^3}{1680}+\frac {t^2}{40}+\frac {t}{2}+1\right ) \]