13.14.23 problem 23

Internal problem ID [2463]
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 : 23
Date solved : Monday, January 27, 2025 at 05:53:23 AM
CAS classification : [_Lienard]

\begin{align*} t^{2} y^{\prime \prime }+t y^{\prime }+t^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 32

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

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1-\frac {1}{4} t^{2}+\frac {1}{64} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {1}{4} t^{2}-\frac {3}{128} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 60

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

\[ y(t)\to c_1 \left (\frac {t^4}{64}-\frac {t^2}{4}+1\right )+c_2 \left (-\frac {3 t^4}{128}+\frac {t^2}{4}+\left (\frac {t^4}{64}-\frac {t^2}{4}+1\right ) \log (t)\right ) \]