14.14.22 problem 22

Internal problem ID [2659]
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 : 22
Date solved : Monday, January 27, 2025 at 06:05:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 y t&=0 \end{align*}

Using series method with expansion around

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

✗ Solution by Maple

Order:=6; 
dsolve(t^2*diff(y(t),t$2)+(1-t^2)*diff(y(t),t)+4*t*y(t)=0,y(t),type='series',t=0);
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 77

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

\[ y(t)\to c_2 e^{\frac {1}{t}} \left (\frac {39467 t^5}{24}+\frac {2135 t^4}{8}+\frac {305 t^3}{6}+\frac {23 t^2}{2}+3 t+1\right ) t^2+c_1 \left (\frac {32 t^5}{15}-t^4+\frac {4 t^3}{3}-2 t^2+1\right ) \]