89.23.20 problem 20

Internal problem ID [24824]
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 : 20
Date solved : Thursday, October 02, 2025 at 10:48:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+13 y&=24 \,{\mathrm e}^{2 x} \cos \left (3 x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 35

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+13*y(x) = 24*exp(2*x)*cos(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{2 x} \left (3 c_1 +4\right ) \cos \left (3 x \right )}{3}+4 \sin \left (3 x \right ) {\mathrm e}^{2 x} \left (x +\frac {c_2}{4}\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 38

ode=D[y[x],{x,2}]-4*D[y[x],x]+13*y[x]==24*Exp[2*x]*Cos[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} e^{2 x} ((2+3 c_2) \cos (3 x)+3 (4 x+c_1) \sin (3 x)) \end{align*}

✓ Sympy. Time used: 0.197 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(13*y(x) - 24*exp(2*x)*cos(3*x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{2} \cos {\left (3 x \right )} + \left (C_{1} + 4 x\right ) \sin {\left (3 x \right )}\right ) e^{2 x} \]