6.19.70 problem section 9.3, problem 70

Internal problem ID [2217]
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 70
Date solved : Tuesday, September 30, 2025 at 05:25:07 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y&=-{\mathrm e}^{-x} \left (4-8 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 22

ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)-diff(y(x),x)+y(x) = -exp(-x)*(4-8*x); 
ic:=[y(0) = 2, D(y)(0) = 0, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = x^{2} {\mathrm e}^{-x}+2 \cosh \left (x \right )-2 x \sinh \left (x \right ) \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 27

ode=D[y[x],{x,3}]-1*D[y[x],{x,2}]-1*D[y[x],x]+1*y[x]==-Exp[-x]*(4-8*x); 
ic={y[0]==2,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} \left (x^2+x-e^{2 x} (x-1)+1\right ) \end{align*}

✓ Sympy. Time used: 0.198 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4 - 8*x)*exp(-x) + y(x) - Derivative(y(x), x) - Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (1 - x\right ) e^{x} + \left (x^{2} + x + 1\right ) e^{- x} \]