76.23.3 problem 4

Internal problem ID [17715]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.1 (Definitions and examples). Problems at page 388
Problem number : 4
Date solved : Thursday, March 13, 2025 at 10:47:56 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime }+3 y&=t \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 84

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+6*diff(diff(diff(y(t),t),t),t)+3*y(t) = t; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {t}{3}+c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{3}+3, \operatorname {index} =1\right ) t}+c_{2} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{3}+3, \operatorname {index} =2\right ) t}+c_{3} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{3}+3, \operatorname {index} =3\right ) t}+c_4 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{3}+3, \operatorname {index} =4\right ) t} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 103

ode=D[y[t],{t,4}]+6*D[y[t],{t,3}]+3*y[t]==t; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^4+6 \text {$\#$1}^3+3\&,3\right ]\right )+c_4 \exp \left (t \text {Root}\left [\text {$\#$1}^4+6 \text {$\#$1}^3+3\&,4\right ]\right )+c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^4+6 \text {$\#$1}^3+3\&,2\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^4+6 \text {$\#$1}^3+3\&,1\right ]\right )+\frac {t}{3} \]

✗ Sympy

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

NotImplementedError : Cannot find 4 solutions to the homogeneous equation necessary to apply undetermined coefficients to -t + 3*y(t) + 6*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)) (number of terms != order)