6.19.39 problem section 9.3, problem 39

Internal problem ID [2186]
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 39
Date solved : Tuesday, September 30, 2025 at 05:24:44 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+15 y&={\mathrm e}^{2 x} \left (15 x \cos \left (2 x \right )+32 \sin \left (2 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 37

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-8*diff(diff(diff(y(x),x),x),x)+24*diff(diff(y(x),x),x)-32*diff(y(x),x)+15*y(x) = exp(2*x)*(15*x*cos(2*x)+32*sin(2*x)); 
dsolve(ode,y(x), singsol=all);
 

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

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

ode=1*D[y[x],{x,4}]-8*D[y[x],{x,3}]+24*D[y[x],{x,2}]-32*D[y[x],x]+15*y[x]==Exp[2*x]*(15*x*Cos[2*x]+32*Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (e^x x \cos (2 x)+c_4 e^{2 x}+c_2 e^x \cos (x)+c_1 e^x \sin (x)+c_3\right ) \end{align*}

✓ Sympy. Time used: 0.407 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-15*x*cos(2*x) - 32*sin(2*x))*exp(2*x) + 15*y(x) - 32*Derivative(y(x), x) + 24*Derivative(y(x), (x, 2)) - 8*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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