6.19.46 problem section 9.3, problem 46

Internal problem ID [2193]
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 46
Date solved : Tuesday, September 30, 2025 at 05:24:49 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+32 y^{\prime \prime }-64 y^{\prime }+64 y&={\mathrm e}^{2 x} \left (\cos \left (2 x \right )-\sin \left (2 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 52

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-8*diff(diff(diff(y(x),x),x),x)+32*diff(diff(y(x),x),x)-64*diff(y(x),x)+64*y(x) = exp(2*x)*(cos(2*x)-sin(2*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{2 x} \left (\left (x^{2}+\left (-32 c_3 -1\right ) x -32 c_1 -\frac {3}{8}\right ) \cos \left (2 x \right )-\sin \left (2 x \right ) \left (x^{2}+\left (32 c_4 +1\right ) x +32 c_2 -\frac {3}{8}\right )\right )}{32} \]

✓ Mathematica. Time used: 0.128 (sec). Leaf size: 65

ode=1*D[y[x],{x,4}]-8*D[y[x],{x,3}]+32*D[y[x],{x,2}]-64*D[y[x],x]+64*y[x]==Exp[2*x]*(Cos[2*x]-Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{256} e^{2 x} \left (\left (-8 x^2+4 (1+64 c_4) x+5+256 c_3\right ) \cos (2 x)+\left (8 x^2+8 (1+32 c_2) x-1+256 c_1\right ) \sin (2 x)\right ) \end{align*}

✓ Sympy. Time used: 0.441 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((sin(2*x) - cos(2*x))*exp(2*x) + 64*y(x) - 64*Derivative(y(x), x) + 32*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)
 

\[ y{\left (x \right )} = \left (- \frac {\sqrt {2} x^{2} \cos {\left (2 x + \frac {\pi }{4} \right )}}{32} + \left (C_{1} + C_{2} x\right ) \sin {\left (2 x \right )} + \left (C_{3} + C_{4} x\right ) \cos {\left (2 x \right )}\right ) e^{2 x} \]