82.30.4 problem Ex. 4

Internal problem ID [18817]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. problems at page 77
Problem number : Ex. 4
Date solved : Monday, March 31, 2025 at 06:15:39 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 58

ode:=diff(diff(diff(y(x),x),x),x)+y(x) = sin(3*x)-cos(1/2*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_3 \,{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+c_1 \,{\mathrm e}^{-x}+\frac {27 \cos \left (3 x \right )}{730}+\frac {\sin \left (3 x \right )}{730}-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}-\frac {1}{2} \]

✓ Mathematica. Time used: 1.739 (sec). Leaf size: 87

ode=D[y[x],{x,3}]+y[x]==Sin[3*x]-Cos[1/2*x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\sin (x)}{4}+\frac {1}{730} \sin (3 x)-\frac {\cos (x)}{4}+\frac {27}{730} \cos (3 x)+c_1 e^{-x}+c_3 e^{x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_2 e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )-\frac {1}{2} \]

✓ Sympy. Time used: 27.454 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = C_{3} e^{- x} + \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{\frac {x}{2}} + \frac {\sin {\left (3 x \right )}}{730} + \frac {27 \cos {\left (3 x \right )}}{730} - \frac {\sqrt {2} \cos {\left (x + \frac {\pi }{4} \right )}}{4} - \frac {1}{2} \]