89.18.19 problem 19

Internal problem ID [24719]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Oral Exercises at page 146
Problem number : 19
Date solved : Thursday, October 02, 2025 at 10:47:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }-y&=x +{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

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

\[ y = {\mathrm e}^{\frac {x}{2}} c_2 +{\mathrm e}^{-\frac {x}{2}} c_1 -x +\frac {{\mathrm e}^{x}}{3} \]

✓ Mathematica. Time used: 0.15 (sec). Leaf size: 36

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

\begin{align*} y(x)&\to -x+\frac {e^x}{3}+c_1 e^{x/2}+c_2 e^{-x/2} \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 22

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

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