13.14.5 problem 5

Internal problem ID [2445]
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 : 5
Date solved : Monday, January 27, 2025 at 05:53:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -1 \end{align*}

✗ Solution by Maple

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

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.086 (sec). Leaf size: 111

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

\[ y(t)\to c_2 e^{\frac {1}{2 (t+1)}} \left (\frac {516353141702117 (t+1)^5}{33443020800}+\frac {53349163853 (t+1)^4}{39813120}+\frac {58276991 (t+1)^3}{414720}+\frac {21397 (t+1)^2}{1152}+\frac {79 (t+1)}{24}+1\right ) (t+1)^{7/4}+c_1 \left (\frac {53}{5} (t+1)^5-\frac {25}{12} (t+1)^4+\frac {2}{3} (t+1)^3-\frac {1}{2} (t+1)^2+1\right ) \]