Internal problem ID [1817]
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: 25.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
Solve \begin {gather*} \boxed {t y^{\prime \prime }+\left (-t +1\right ) y^{\prime }+\lambda 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: 309
Order:=6; dsolve(t*diff(y(t),t$2)+(1-t)*diff(y(t),t)+lambda*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (\left (2 \lambda +1\right ) t +\left (\frac {1}{4} \lambda +\frac {1}{4}-\frac {3}{4} \lambda ^{2}\right ) t^{2}+\left (-\frac {2}{9} \lambda ^{2}+\frac {1}{27} \lambda +\frac {1}{18}+\frac {11}{108} \lambda ^{3}\right ) t^{3}+\left (\frac {7}{192} \lambda ^{3}-\frac {167}{3456} \lambda ^{2}+\frac {1}{192} \lambda +\frac {1}{96}-\frac {25}{3456} \lambda ^{4}\right ) t^{4}+\left (\frac {1}{1500} \lambda +\frac {137}{432000} \lambda ^{5}-\frac {37}{4320} \lambda ^{2}+\frac {719}{86400} \lambda ^{3}-\frac {61}{21600} \lambda ^{4}+\frac {1}{600}\right ) t^{5}+\mathrm {O}\left (t^{6}\right )\right ) c_{2}+\left (1-\lambda t +\frac {1}{4} \left (-1+\lambda \right ) \lambda t^{2}-\frac {1}{36} \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda t^{3}+\frac {1}{576} \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda t^{4}-\frac {1}{14400} \left (\lambda -4\right ) \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda t^{5}+\mathrm {O}\left (t^{6}\right )\right ) \left (c_{2} \ln \relax (t )+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 415
AsymptoticDSolveValue[t*y''[t]+(1-t)*y'[t]+\[Lambda]*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda t^3+\frac {1}{4} (\lambda -1) \lambda t^2-\lambda t+1\right )+c_2 \left (\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) t^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -2) \lambda t^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -1) \lambda t^5}{14400}+\frac {(\lambda -4) (\lambda -2) (\lambda -1) \lambda t^5}{14400}+\frac {137 (\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{432000}+\frac {(\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{14400}-\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) t^4-\frac {1}{576} (\lambda -3) (\lambda -2) \lambda t^4-\frac {1}{576} (\lambda -3) (\lambda -1) \lambda t^4-\frac {25 (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^4}{3456}-\frac {1}{576} (\lambda -2) (\lambda -1) \lambda t^4+\frac {1}{36} (\lambda -2) (\lambda -1) t^3+\frac {1}{36} (\lambda -2) \lambda t^3+\frac {11}{108} (\lambda -2) (\lambda -1) \lambda t^3+\frac {1}{36} (\lambda -1) \lambda t^3-\frac {1}{4} (\lambda -1) t^2-\frac {3}{4} (\lambda -1) \lambda t^2-\frac {\lambda t^2}{4}+\left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda t^3+\frac {1}{4} (\lambda -1) \lambda t^2-\lambda t+1\right ) \log (t)+2 \lambda t+t\right ) \]