6.19.38 problem section 9.3, problem 38

Internal problem ID [2185]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 38
Date solved : Tuesday, September 30, 2025 at 05:24:43 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+2 y^{\prime \prime }+2 y^{\prime }-4 y&={\mathrm e}^{x} \left (2 \cos \left (2 x \right )-\sin \left (2 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 41

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-3*diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x)+2*diff(y(x),x)-4*y(x) = exp(x)*(2*cos(2*x)-sin(2*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{-x}+c_2 \,{\mathrm e}^{2 x}+\left (\frac {\cos \left (x \right )^{2}}{6}+\left (c_3 -\frac {\sin \left (x \right )}{6}\right ) \cos \left (x \right )+c_4 \sin \left (x \right )-\frac {1}{12}\right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 56

ode=1*D[y[x],{x,4}]-3*D[y[x],{x,3}]+2*D[y[x],{x,2}]+2*D[y[x],x]-4*y[x]==Exp[x]*(2*Cos[2*x]-Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_3 e^{-x}+c_4 e^{2 x}+\frac {1}{12} e^x (\cos (2 x)-\sin (2 x))+c_2 e^x \cos (x)+c_1 e^x \sin (x) \end{align*}

✓ Sympy. Time used: 0.299 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((sin(2*x) - 2*cos(2*x))*exp(x) - 4*y(x) + 2*Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} e^{- x} + C_{4} e^{2 x} + \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )} - \frac {\sin {\left (2 x \right )}}{12} + \frac {\cos {\left (2 x \right )}}{12}\right ) e^{x} \]