13.14.16 problem 16

Internal problem ID [2456]
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 : 16
Date solved : Monday, January 27, 2025 at 05:53:13 AM
CAS classification : [_Laguerre, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t^{2} y^{\prime \prime }+\left (-t^{2}+3 t \right ) y^{\prime }-y t&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 44

Order:=6; 
dsolve(t^2*diff(y(t),t$2)+(3*t-t^2)*diff(y(t),t)-t*y(t)=0,y(t),type='series',t=0);
 

\[ y = c_1 \left (1+\frac {1}{3} t +\frac {1}{12} t^{2}+\frac {1}{60} t^{3}+\frac {1}{360} t^{4}+\frac {1}{2520} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\frac {c_2 \left (-2-2 t -t^{2}-\frac {1}{3} t^{3}-\frac {1}{12} t^{4}-\frac {1}{60} t^{5}+\operatorname {O}\left (t^{6}\right )\right )}{t^{2}} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 60

AsymptoticDSolveValue[t^2*D[y[t],{t,2}]+(3*t-t^2)*D[y[t],t]-t*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_1 \left (\frac {t^2}{24}+\frac {1}{t^2}+\frac {t}{6}+\frac {1}{t}+\frac {1}{2}\right )+c_2 \left (\frac {t^4}{360}+\frac {t^3}{60}+\frac {t^2}{12}+\frac {t}{3}+1\right ) \]