14.12.10 problem 10

Internal problem ID [2620]
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 : 10
Date solved : Sunday, March 30, 2025 at 12:11:45 AM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 81

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

\[ y = \left (1-\frac {\alpha \left (\alpha +1\right ) t^{2}}{2}+\frac {\alpha \left (\alpha ^{3}+2 \alpha ^{2}-5 \alpha -6\right ) t^{4}}{24}\right ) y \left (0\right )+\left (t -\frac {\left (\alpha ^{2}+\alpha -2\right ) t^{3}}{6}+\frac {\left (\alpha ^{4}+2 \alpha ^{3}-13 \alpha ^{2}-14 \alpha +24\right ) t^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 127

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

\[ y(t)\to c_2 \left (\frac {1}{60} \left (-\alpha ^2-\alpha \right ) t^5-\frac {1}{120} \left (-\alpha ^2-\alpha \right ) \left (\alpha ^2+\alpha \right ) t^5-\frac {1}{10} \left (\alpha ^2+\alpha \right ) t^5+\frac {t^5}{5}-\frac {1}{6} \left (\alpha ^2+\alpha \right ) t^3+\frac {t^3}{3}+t\right )+c_1 \left (\frac {1}{24} \left (\alpha ^2+\alpha \right )^2 t^4-\frac {1}{4} \left (\alpha ^2+\alpha \right ) t^4-\frac {1}{2} \left (\alpha ^2+\alpha \right ) t^2+1\right ) \]

✓ Sympy. Time used: 1.095 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
Alpha = symbols("Alpha") 
y = Function("y") 
ode = Eq(Alpha*(Alpha + 1)*y(t) - 2*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)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {\mathrm {A}^{4} t^{4}}{24} + \frac {\mathrm {A}^{3} t^{4}}{12} - \frac {5 \mathrm {A}^{2} t^{4}}{24} - \frac {\mathrm {A}^{2} t^{2}}{2} - \frac {\mathrm {A} t^{4}}{4} - \frac {\mathrm {A} t^{2}}{2} + 1\right ) + C_{1} t \left (- \frac {\mathrm {A}^{2} t^{2}}{6} - \frac {\mathrm {A} t^{2}}{6} + \frac {t^{2}}{3} + 1\right ) + O\left (t^{6}\right ) \]