14.12.11 problem 11

Internal problem ID [2621]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8. Series solutions. Excercises page 197
Problem number : 11
Date solved : Sunday, March 30, 2025 at 12:11:46 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 65

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

\[ y = \left (1-\frac {\alpha ^{2} t^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) t^{4}}{24}\right ) y \left (0\right )+\left (t -\frac {\left (\alpha ^{2}-1\right ) t^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) t^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 88

ode=(1-t^2)*D[y[t],{t,2}]-t*D[y[t],t]+\[Alpha]^2*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_2 \left (\frac {\alpha ^4 t^5}{120}-\frac {\alpha ^2 t^5}{12}+\frac {3 t^5}{40}-\frac {\alpha ^2 t^3}{6}+\frac {t^3}{6}+t\right )+c_1 \left (\frac {\alpha ^4 t^4}{24}-\frac {\alpha ^2 t^4}{6}-\frac {\alpha ^2 t^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.931 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
Alpha = symbols("Alpha") 
y = Function("y") 
ode = Eq(Alpha**2*y(t) - t*Derivative(y(t), t) + (1 - t**2)*Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {\mathrm {A}^{4} t^{4}}{24} - \frac {\mathrm {A}^{2} t^{4}}{6} - \frac {\mathrm {A}^{2} t^{2}}{2} + 1\right ) + C_{1} t \left (- \frac {\mathrm {A}^{2} t^{2}}{6} + \frac {t^{2}}{6} + 1\right ) + O\left (t^{6}\right ) \]