75.16.59 problem 532

Internal problem ID [16961]
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 : 532
Date solved : Monday, March 31, 2025 at 03:36:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 42

ode:=diff(diff(y(x),x),x)+a^2*y(x) = 2*cos(m*x)+3*sin(m*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (a x \right ) c_2 +\cos \left (a x \right ) c_1 +\frac {2 \cos \left (m x \right )+3 \sin \left (m x \right )}{a^{2}-m^{2}} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 45

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

\[ y(x)\to \frac {3 \sin (m x)+2 \cos (m x)}{a^2-m^2}+c_1 \cos (a x)+c_2 \sin (a x) \]

✓ Sympy. Time used: 0.160 (sec). Leaf size: 46

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

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