74.12.53 problem 61 (a)

Internal problem ID [16281]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 61 (a)
Date solved : Thursday, March 13, 2025 at 08:09:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }-4 t y^{\prime }+\left (t^{2}+6\right ) y&=t^{3}+2 t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.102 (sec). Leaf size: 18

ode:=t^2*diff(diff(y(t),t),t)-4*t*diff(y(t),t)+(t^2+6)*y(t) = t^3+2*t; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = t \left (\sin \left (t \right ) t c_{2} +\cos \left (t \right ) t c_{1} +1\right ) \]

✓ Mathematica. Time used: 0.329 (sec). Leaf size: 37

ode=t^2*D[y[t],{t,2}]-4*t*D[y[t],t]+(t^2+6)*y[t]==t^3+2*t; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to t+t^2 \left (c_1 e^{-i t}-\frac {1}{2} i c_2 e^{i t}\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**3 + t**2*Derivative(y(t), (t, 2)) - 4*t*Derivative(y(t), t) - 2*t + (t**2 + 6)*y(t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(t), t) - (t*(-t**2 + t*y(t) + t*Derivative(y(t), (t, 2)) - 2) + 6*y(t))/(4*t) cannot be solved by the factorable group method