14.12.9 problem 9

Internal problem ID [2619]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8. Series solutions. Excercises page 197
Problem number : 9
Date solved : Monday, January 27, 2025 at 06:04:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 t y^{\prime }+\lambda y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 53

Order:=6; 
dsolve(diff(y(t),t$2)-2*t*diff(y(t),t)+lambda*y(t)=0,y(t),type='series',t=0);
 

\[ y = \left (1-\frac {\lambda \,t^{2}}{2}+\frac {\lambda \left (\lambda -4\right ) t^{4}}{24}\right ) y \left (0\right )+\left (t -\frac {\left (\lambda -2\right ) t^{3}}{6}+\frac {\left (\lambda -2\right ) \left (-6+\lambda \right ) t^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 80

AsymptoticDSolveValue[D[y[t],{t,2}]-2*t*D[y[t],t]+\[Lambda]*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_2 \left (\frac {\lambda ^2 t^5}{120}-\frac {\lambda t^5}{15}+\frac {t^5}{10}-\frac {\lambda t^3}{6}+\frac {t^3}{3}+t\right )+c_1 \left (\frac {\lambda ^2 t^4}{24}-\frac {\lambda t^4}{6}-\frac {\lambda t^2}{2}+1\right ) \]