13.14.15 problem 15

Internal problem ID [2455]
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 : 15
Date solved : Monday, January 27, 2025 at 05:53:12 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 32

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

\[ y = c_1 \,t^{3} \left (1+\frac {1}{5} t^{2}+\frac {1}{35} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (12+6 t^{2}+\frac {3}{2} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 44

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

\[ y(t)\to c_1 \left (\frac {t^4}{8}+\frac {t^2}{2}+1\right )+c_2 \left (\frac {t^7}{35}+\frac {t^5}{5}+t^3\right ) \]