89.23.15 problem 15

Internal problem ID [24819]
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. Miscellaneous Exercises at page 162
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:48:18 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=8 x +8 \sin \left (4 x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)+16*y(x) = 8*x+8*sin(4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 -x \right ) \cos \left (4 x \right )+\sin \left (4 x \right ) c_2 +\frac {x}{2} \]

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 29

ode=D[y[x],{x,2}]+16*y[x]==8*(x+Sin[4*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x}{2}+(-x+c_1) \cos (4 x)+c_2 \sin (4 x) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x + 16*y(x) - 8*sin(4*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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