14.14.18 problem 18

Internal problem ID [2655]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.2, Regular singular points, the method of Frobenius. Excercises page 216
Problem number : 18
Date solved : Monday, January 27, 2025 at 06:05:29 AM
CAS classification : [_Laguerre]

\begin{align*} t y^{\prime \prime }-\left (4+t \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 1.070 (sec). Leaf size: 40

Order:=6; 
dsolve(t*diff(y(t),t$2)-(4+t)*diff(y(t),t)+2*y(t)=0,y(t),type='series',t=0);
 

\[ y = c_1 \,t^{5} \left (1+\frac {1}{2} t +\frac {1}{7} t^{2}+\frac {5}{168} t^{3}+\frac {5}{1008} t^{4}+\frac {1}{1440} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (2880+1440 t +240 t^{2}+4 t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 56

AsymptoticDSolveValue[t*D[y[t],{t,2}]-(4+t)*D[y[t],t]+2*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_1 \left (\frac {t^2}{12}+\frac {t}{2}+1\right )+c_2 \left (\frac {5 t^9}{1008}+\frac {5 t^8}{168}+\frac {t^7}{7}+\frac {t^6}{2}+t^5\right ) \]