65.11.6 problem 6

Internal problem ID [15802]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.4, page 218
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:28:10 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime \prime }-y^{\prime }-y&=\left (2 x^{2}+4 x +8\right ) \cos \left (x \right )+\left (6 x^{2}+8 x +12\right ) \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 43

ode:=diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x)-diff(y(x),x)-y(x) = (2*x^2+4*x+8)*cos(x)+(6*x^2+8*x+12)*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_3 x +c_2 \right ) {\mathrm e}^{-x}+\left (x^{2}-6 x -2\right ) \cos \left (x \right )+\left (-2 x^{2}-4 x +1\right ) \sin \left (x \right )+c_1 \,{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 55

ode=D[y[x],{x,3}]+D[y[x],{x,2}]-D[y[x],x]-y[x]==(2*x^2+4*x+8)*Cos[x]+(6*x^2+8*x+12)*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (x^2-6 x-2\right ) \cos (x)+e^{-x} \left (-e^x \left (2 x^2+4 x-1\right ) \sin (x)+c_2 x+c_3 e^{2 x}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.278 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-2*x**2 - 4*x - 8)*cos(x) - (6*x**2 + 8*x + 12)*sin(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)
 

\[ y{\left (x \right )} = C_{3} e^{x} - 2 x^{2} \sin {\left (x \right )} + x^{2} \cos {\left (x \right )} - 4 x \sin {\left (x \right )} - 6 x \cos {\left (x \right )} + \left (C_{1} + C_{2} x\right ) e^{- x} + \sin {\left (x \right )} - 2 \cos {\left (x \right )} \]