87.8.5 problem 5

Internal problem ID [23359]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 65
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:40:27 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\left (5\right )}-2 y^{\prime \prime \prime \prime }+y&=2 x^{2}+3 \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 123

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-2*diff(diff(diff(diff(y(x),x),x),x),x)+y(x) = 2*x^2+3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 x^{2}+3+c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_Z}^{2}-\textit {\_Z} -1, \operatorname {index} =1\right ) x}+c_3 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_Z}^{2}-\textit {\_Z} -1, \operatorname {index} =2\right ) x}+c_4 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_Z}^{2}-\textit {\_Z} -1, \operatorname {index} =3\right ) x}+c_5 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_Z}^{2}-\textit {\_Z} -1, \operatorname {index} =4\right ) x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 150

ode=D[y[x],{x,5}]-2*D[y[x],{x,4}]+y[x]==2*x^2+3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^4-\text {$\#$1}^3-\text {$\#$1}^2-\text {$\#$1}-1\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^4-\text {$\#$1}^3-\text {$\#$1}^2-\text {$\#$1}-1\&,4\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^4-\text {$\#$1}^3-\text {$\#$1}^2-\text {$\#$1}-1\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^4-\text {$\#$1}^3-\text {$\#$1}^2-\text {$\#$1}-1\&,2\right ]\right )+2 x^2+c_5 e^x+3 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2 + y(x) - 2*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)) - 3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Cannot find 5 solutions to the homogeneous equation necessary to apply undeter