89.16.12 problem 12

Internal problem ID [24662]
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 : 12
Date solved : Thursday, October 02, 2025 at 10:46:49 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+3 y&=2 \cos \left (x \right )+4 \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 17

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+3*y(x) = 2*cos(x)+4*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{3 x} c_2 +{\mathrm e}^{x} c_1 +\cos \left (x \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 22

ode=D[y[x],{x,2}]-4*D[y[x],x]+3*y[x]== 2*Cos[x] +4*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos (x)+c_1 e^x+c_2 e^{3 x} \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*y(x) - 4*sin(x) - 2*cos(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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