89.21.6 problem 8

Internal problem ID [24778]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 154
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:47:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }-y&={\mathrm e}^{\frac {x}{2}}+12 \,{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

ode:=4*diff(diff(y(x),x),x)-y(x) = exp(1/2*x)+12*exp(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 39

ode=4*D[y[x],{x,2}]-y[x]== Exp[x/2]+12*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.078 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) - exp(x/2) - 12*exp(x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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