67.15.67 problem 22.13 (e)

Internal problem ID [16785]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 22. Method of undetermined coefficients. Additional exercises page 412
Problem number : 22.13 (e)
Date solved : Thursday, October 02, 2025 at 01:38:49 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 42

ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)+diff(y(x),x)-y(x) = 3*x*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-3 x^{2}+8 c_3 +3 x +9\right ) \sin \left (x \right )}{8}+\frac {\left (-3 x^{2}+8 c_1 -9 x \right ) \cos \left (x \right )}{8}+c_2 \,{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 102

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

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

✓ Sympy. Time used: 0.167 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{3} e^{x} + \left (C_{1} - \frac {3 x^{2}}{8} - \frac {9 x}{8}\right ) \cos {\left (x \right )} + \left (C_{2} - \frac {3 x^{2}}{8} + \frac {3 x}{8}\right ) \sin {\left (x \right )} \]