83.14.2 problem 2

Internal problem ID [19110]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter III. Ordinary linear differential equations with constant coefficients. Exercise III (F) at page 42
Problem number : 2
Date solved : Monday, March 31, 2025 at 06:48:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 81

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

\[ y = \frac {\left (4 c_1 \,a^{4}+\left (-4 c_1 \,b^{2}+1\right ) a^{2}-b^{2}\right ) \cos \left (a x \right )+4 a \left (\left (a c_2 +\frac {x}{2}\right ) \left (a +b \right ) \left (a -b \right ) \sin \left (a x \right )+a \cos \left (b x \right )\right )}{4 a^{4}-4 a^{2} b^{2}} \]

✓ Mathematica. Time used: 0.571 (sec). Leaf size: 79

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

\[ y(x)\to \frac {\left (a^2-b^2\right ) \left (1+2 a^2 c_1\right ) \cos (a x)+a \left (2 a \cos (b x)+\left (a^2-b^2\right ) (x+2 a c_2) \sin (a x)\right )}{2 a^2 (a-b) (a+b)} \]

✓ Sympy. Time used: 0.157 (sec). Leaf size: 41

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

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