Internal problem ID [1796]
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: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left ({\mathrm e}^{t}-1\right ) y^{\prime \prime }+{\mathrm e}^{t} y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 59
Order:=6; dsolve((exp(t)-1)*diff(y(t),t$2)+exp(t)*diff(y(t),t)+y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (c_{2} \ln \relax (t )+c_{1}\right ) \left (1-t +\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (\frac {3}{2} t -\frac {23}{24} t^{2}+\frac {3}{8} t^{3}-\frac {301}{2880} t^{4}+\frac {13}{576} t^{5}+\mathrm {O}\left (t^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 113
AsymptoticDSolveValue[(Exp[t]-1)*y''[t]+Exp[t]*y'[t]+y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (-\frac {t^5}{120}+\frac {t^4}{24}-\frac {t^3}{6}+\frac {t^2}{2}-t+1\right )+c_2 \left (\frac {13 t^5}{576}-\frac {301 t^4}{2880}+\frac {3 t^3}{8}-\frac {23 t^2}{24}+\left (-\frac {t^5}{120}+\frac {t^4}{24}-\frac {t^3}{6}+\frac {t^2}{2}-t+1\right ) \log (t)+\frac {3 t}{2}\right ) \]