75.16.73 problem 546

Internal problem ID [16975]
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 : 546
Date solved : Monday, March 31, 2025 at 03:36:55 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y&=\sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 43

ode:=diff(diff(diff(y(x),x),x),x)-y(x) = sin(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.235 (sec). Leaf size: 224

ode=D[y[x],{x,3}]-y[x]==Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x/2} \left (e^{3 x/2} \int _1^x\frac {1}{3} e^{-K[1]} \sin (K[1])dK[1]+\cos \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x-\frac {e^{\frac {K[2]}{2}} \sin (K[2]) \left (\sqrt {3} \cos \left (\frac {1}{2} \sqrt {3} K[2]\right )-3 \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )\right )}{3 \sqrt {3}}dK[2]+\sin \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x-\frac {e^{\frac {K[3]}{2}} \sin (K[3]) \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[3]\right )+\sqrt {3} \sin \left (\frac {1}{2} \sqrt {3} K[3]\right )\right )}{3 \sqrt {3}}dK[3]+c_1 e^{3 x/2}+c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.150 (sec). Leaf size: 46

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

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