14.12.3 problem 3

Internal problem ID [2613]
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 : 3
Date solved : Tuesday, March 04, 2025 at 02:32:13 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 34

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

\[ y = \left (1+\frac {3}{4} t^{2}+\frac {3}{32} t^{4}\right ) y \left (0\right )+\left (\frac {1}{3} t^{3}+t \right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 35

ode=(2+t^2)*D[y[t],{t,2}]-t*D[y[t],t]-3*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_2 \left (\frac {t^3}{3}+t\right )+c_1 \left (\frac {3 t^4}{32}+\frac {3 t^2}{4}+1\right ) \]

✓ Sympy. Time used: 0.827 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*Derivative(y(t), t) + (t**2 + 2)*Derivative(y(t), (t, 2)) - 3*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {3 t^{4}}{32} + \frac {3 t^{2}}{4} + 1\right ) + C_{1} t \left (\frac {t^{2}}{3} + 1\right ) + O\left (t^{6}\right ) \]