13.15.3 problem 3

Internal problem ID [2470]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.3, The method of Frobenius. Equal roots, and roots differering by an integer. Page 223
Problem number : 3
Date solved : Monday, January 27, 2025 at 05:53:31 AM
CAS classification : [_Bessel]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 46

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

\[ y = \frac {c_1 \,t^{2} \left (1-\frac {1}{8} t^{2}+\frac {1}{192} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (t^{2}-\frac {1}{8} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+\left (-2+\frac {3}{32} t^{4}+\operatorname {O}\left (t^{6}\right )\right )\right )}{t} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 58

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

\[ y(t)\to c_2 \left (\frac {t^5}{192}-\frac {t^3}{8}+t\right )+c_1 \left (\frac {1}{16} t \left (t^2-8\right ) \log (t)-\frac {5 t^4-16 t^2-64}{64 t}\right ) \]