59.1.211 problem 214

Internal problem ID [9383]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 214
Date solved : Sunday, March 30, 2025 at 02:33:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 t y&=0 \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 21

ode:=t*diff(diff(y(t),t),t)+(-t^2+1)*diff(y(t),t)+4*t*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (t^{4}-8 t^{2}+8\right ) \left (c_1 +2 c_2 \right )}{8} \]

✓ Mathematica. Time used: 0.622 (sec). Leaf size: 65

ode=t*D[y[t],{t,2}]+(1-t^2)*D[y[t],t]+4*t*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \sqrt {e} \left (t^4-8 t^2+8\right ) \left (c_2 \int _1^t\frac {e^{\frac {K[1]^2}{2}-1}}{K[1] \left (K[1]^4-8 K[1]^2+8\right )^2}dK[1]+c_1\right ) \]

✓ Sympy. Time used: 0.775 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*t*y(t) + t*Derivative(y(t), (t, 2)) + (1 - t**2)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} \left (\frac {t^{4}}{8} - t^{2} + 1\right ) + O\left (t^{6}\right ) \]