Internal problem ID [1820]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {t y^{\prime \prime }+y^{\prime }-4 y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 59
Order:=6; dsolve(t*diff(y(t),t$2)+diff(y(t),t)-4*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (c_{2} \ln \relax (t )+c_{1}\right ) \left (1+4 t +4 t^{2}+\frac {16}{9} t^{3}+\frac {4}{9} t^{4}+\frac {16}{225} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (\left (-8\right ) t -12 t^{2}-\frac {176}{27} t^{3}-\frac {50}{27} t^{4}-\frac {1096}{3375} t^{5}+\mathrm {O}\left (t^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 105
AsymptoticDSolveValue[t*y''[t]+y'[t]-4*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {16 t^5}{225}+\frac {4 t^4}{9}+\frac {16 t^3}{9}+4 t^2+4 t+1\right )+c_2 \left (-\frac {1096 t^5}{3375}-\frac {50 t^4}{27}-\frac {176 t^3}{27}-12 t^2+\left (\frac {16 t^5}{225}+\frac {4 t^4}{9}+\frac {16 t^3}{9}+4 t^2+4 t+1\right ) \log (t)-8 t\right ) \]