34.11.5 problem 30

Internal problem ID [8026]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 16. Linear equations with constant coefficients (Short methods). Supplemetary problems. Page 107
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 05:14:23 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 45

ode:=diff(diff(diff(y(x),x),x),x)+y(x) = cos(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.208 (sec). Leaf size: 232

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

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

✓ Sympy. Time used: 0.087 (sec). Leaf size: 46

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

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