6.19.56 problem section 9.3, problem 56

Internal problem ID [2203]
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 56
Date solved : Tuesday, September 30, 2025 at 05:24:58 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+4 y&=2 \,{\mathrm e}^{x} \left (1+x \right )+{\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 50

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = 2*(1+x)*exp(x)+exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (27 x^{2}+\left (486 c_4 +36\right ) x +486 c_2 +18\right ) {\mathrm e}^{-2 x}}{486}+\frac {{\mathrm e}^{x} \left (x^{3}+x^{2}+\left (27 c_3 -2\right ) x +27 c_1 +\frac {10}{9}\right )}{27} \]

✓ Mathematica. Time used: 0.174 (sec). Leaf size: 66

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

\begin{align*} y(x)&\to \frac {1}{54} e^{-2 x} \left (3 x^2+(4+54 c_2) x+2+54 c_1\right )+\frac {1}{243} e^x \left (9 x^3+9 x^2+9 (-2+27 c_4) x+10+243 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.298 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x}{18}\right )\right ) e^{- 2 x} + \left (C_{3} + x \left (C_{4} + \frac {x^{2}}{27} + \frac {x}{27}\right )\right ) e^{x} \]