6.19.18 problem section 9.3, problem 18

Internal problem ID [2165]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 18
Date solved : Tuesday, September 30, 2025 at 05:24:31 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+2 y&={\mathrm e}^{2 x} \left (x^{4}+x +24\right ) \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 87

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-4*diff(diff(diff(y(x),x),x),x)+6*diff(diff(y(x),x),x)-4*diff(y(x),x)+2*y(x) = exp(2*x)*(x^4+x+24); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {\left (\sqrt {2}-2\right ) x}{2}} \left (c_3 \cos \left (\frac {\sqrt {2}\, x}{2}\right )+c_4 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )+{\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \left (c_1 \cos \left (\frac {\sqrt {2}\, x}{2}\right )+c_2 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )+\frac {{\mathrm e}^{2 x} \left (x^{4}-8 x^{3}+12 x^{2}+49 x -62\right )}{2} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 102

ode=D[y[x],{x,4}]-4*D[y[x],{x,3}]+6*D[y[x],{x,2}]-4*D[y[x],x]+2*y[x]==Exp[2*x]*(24+x+x^4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} e^{x-\frac {x}{\sqrt {2}}} \left (e^{\frac {x}{\sqrt {2}}+x} \left (x^4-8 x^3+12 x^2+49 x-62\right )+2 \left (c_4 e^{\sqrt {2} x}+c_2\right ) \cos \left (\frac {x}{\sqrt {2}}\right )+2 \left (c_1 e^{\sqrt {2} x}+c_3\right ) \sin \left (\frac {x}{\sqrt {2}}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x**4 - x - 24)*exp(2*x) + 2*y(x) - 4*Derivative(y(x), x) + 6*Derivative(y(x), (x, 2)) - 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x**4*exp(2*x)/4 + x*exp(2*x)/4 - y(x)/2 + 6*exp(2*x) + Derivative(y(x), x) - 3*Derivative(y(x), (x, 2))/2 + Derivative(y(x), (x, 3)) - Derivative(y(x), (x, 4))/4 cannot be solved by the factorable group method