Internal problem ID [1823]
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: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {t y^{\prime \prime }+3 y^{\prime }-3 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: 62
Order:=6; dsolve(t*diff(y(t),t$2)+3*diff(y(t),t)-3*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \frac {c_{1} \left (1+t +\frac {3}{8} t^{2}+\frac {3}{40} t^{3}+\frac {3}{320} t^{4}+\frac {9}{11200} t^{5}+\mathrm {O}\left (t^{6}\right )\right ) t^{2}+c_{2} \left (\ln \relax (t ) \left (9 t^{2}+9 t^{3}+\frac {27}{8} t^{4}+\frac {27}{40} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (-2+6 t -12 t^{3}-\frac {225}{32} t^{4}-\frac {1413}{800} t^{5}+\mathrm {O}\left (t^{6}\right )\right )\right )}{t^{2}} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 78
AsymptoticDSolveValue[t*y''[t]+3*y'[t]-3*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 \left (\frac {3 t^4}{320}+\frac {3 t^3}{40}+\frac {3 t^2}{8}+t+1\right )+c_1 \left (\frac {279 t^4+528 t^3+144 t^2-192 t+64}{64 t^2}-\frac {9}{16} \left (3 t^2+8 t+8\right ) \log (t)\right ) \]