89.16.9 problem 9

Internal problem ID [24659]
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. Exercises at page 140
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:46:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y&=2+{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

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

\[ y = \frac {\left (4 x +16 c_1 -1\right ) {\mathrm e}^{2 x}}{16}+{\mathrm e}^{-2 x} c_2 -\frac {1}{2} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to e^{2 x} \left (\frac {x}{4}-\frac {1}{16}+c_1\right )+c_2 e^{-2 x}-\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 22

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

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