89.16.13 problem 13

Internal problem ID [24663]
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 : 13
Date solved : Thursday, October 02, 2025 at 10:46:50 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y+2 y^{\prime }+y^{\prime \prime }&=7+75 \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = 7+75*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 x +c_2 \right ) {\mathrm e}^{-x}-12 \cos \left (2 x \right )-9 \sin \left (2 x \right )+7 \]

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

ode=D[y[x],{x,2}]+2*D[y[x],x]+y[x]== 7+75*Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -9 \sin (2 x)-12 \cos (2 x)+e^{-x} \left (7 e^x+c_2 x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.150 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - 75*sin(2*x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 7,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{- x} - 9 \sin {\left (2 x \right )} - 12 \cos {\left (2 x \right )} + 7 \]