12.19.21 problem section 9.3, problem 21

Internal problem ID [2168]
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 21
Date solved : Tuesday, March 04, 2025 at 01:51:03 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }-2 y^{\prime }-y&=3 \,{\mathrm e}^{-\frac {x}{2}} \left (1-6 x \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 50

ode:=2*diff(diff(diff(diff(y(x),x),x),x),x)+diff(diff(diff(y(x),x),x),x)-2*diff(y(x),x)-y(x) = 3*exp(-1/2*x)*(1-6*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.506 (sec). Leaf size: 63

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

\[ y(x)\to e^{-x/2} \left (4 x^2+4 x+c_4 e^{3 x/2}+c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} x}{2}\right )-8+c_3\right ) \]

✓ Sympy. Time used: 0.405 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((18*x - 3)*exp(-x/2) - y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 3)) + 2*Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{4} e^{x} + \left (C_{1} + C_{2} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{3} \cos {\left (\frac {\sqrt {3} x}{2} \right )} + 4 x^{2} + 4 x\right ) e^{- \frac {x}{2}} \]