12.19.65 problem section 9.3, problem 65

Internal problem ID [2212]
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 65
Date solved : Tuesday, March 04, 2025 at 01:51:48 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y&=-{\mathrm e}^{-x} \left (3 x^{2}-9 x +4\right ) \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 40

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-2*diff(diff(y(x),x),x)+y(x) = -exp(-x)*(3*x^2-9*x+4); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-x^{4}+2 x^{3}+16 c_4 x +x^{2}+16 c_2 \right ) {\mathrm e}^{-x}}{16}+{\mathrm e}^{x} \left (x c_3 +c_1 \right ) \]

✓ Mathematica. Time used: 0.146 (sec). Leaf size: 61

ode=D[y[x],{x,4}]-0*D[y[x],{x,3}]-2*D[y[x],{x,2}]+0*D[y[x],x]+1*y[x]==-Exp[-x]*(4-9*x+3*x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{32} e^{-x} \left (-2 x^4+4 x^3+2 x^2+x \left (32 c_4 e^{2 x}-2+32 c_2\right )+32 c_3 e^{2 x}-3+32 c_1\right ) \]

✓ Sympy. Time used: 0.241 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x**2 - 9*x + 4)*exp(-x) + y(x) - 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{x} + \left (C_{3} + x \left (C_{4} - \frac {x^{3}}{16} + \frac {x^{2}}{8} + \frac {x}{16}\right )\right ) e^{- x} \]