69.16.14 problem 487

Internal problem ID [18274]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 487
Date solved : Thursday, October 02, 2025 at 03:10:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&={\mathrm e}^{-3 x} \cos \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+13*y(x) = exp(-3*x)*cos(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\sin \left (2 x \right ) \left (x +4 c_2 \right )+4 \left (c_1 +\frac {1}{8}\right ) \cos \left (2 x \right )\right ) {\mathrm e}^{-3 x}}{4} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 70

ode=D[y[x],{x,2}]+6*D[y[x],x]+13*y[x]==Exp[-3*x]*Cos[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-3 x} \left (\sin (2 x) \int _1^x\frac {1}{2} \cos ^2(2 K[1])dK[1]+\cos (2 x) \int _1^x-\frac {1}{4} \sin (4 K[2])dK[2]+c_2 \cos (2 x)+c_1 \sin (2 x)\right ) \end{align*}

✓ Sympy. Time used: 0.243 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(13*y(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-3*x)*cos(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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