23.3.3 problem 3

Internal problem ID [5717]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 02:02:01 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=\operatorname {c1} \cos \left (a x \right )+\operatorname {c2} \sin \left (b x \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x) = c1*cos(a*x)+c2*sin(b*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\operatorname {c1} \cos \left (a x \right )}{a^{2}}-\frac {\operatorname {c2} \sin \left (b x \right )}{b^{2}}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 32

ode=D[y[x],{x,2}] == c1*Cos[a*x] + c2*Sin[b*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\text {c1} \cos (a x)}{a^2}-\frac {\text {c2} \sin (b x)}{b^2}+c_2 x+c_1 \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c1 = symbols("c1") 
c2 = symbols("c2") 
y = Function("y") 
ode = Eq(-c1*cos(a*x) - c2*sin(b*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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