Internal problem ID [1773]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8, Series solutions. Page 195
Problem number: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 t y^{\prime }+\lambda y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 63
Order:=6; dsolve(diff(y(t),t$2)-2*t*diff(y(t),t)+lambda*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (1-\frac {\lambda \,t^{2}}{2}+\frac {\lambda \left (\lambda -4\right ) t^{4}}{24}\right ) y \relax (0)+\left (t -\frac {\left (\lambda -2\right ) t^{3}}{6}+\frac {\left (\lambda -2\right ) \left (-6+\lambda \right ) t^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (t^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 80
AsymptoticDSolveValue[y''[t]-2*t*y'[t]+\[Lambda]*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 \left (\frac {\lambda ^2 t^5}{120}-\frac {\lambda t^5}{15}+\frac {t^5}{10}-\frac {\lambda t^3}{6}+\frac {t^3}{3}+t\right )+c_1 \left (\frac {\lambda ^2 t^4}{24}-\frac {\lambda t^4}{6}-\frac {\lambda t^2}{2}+1\right ) \]