6.19.68 problem section 9.3, problem 68

Internal problem ID [2215]
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 68
Date solved : Tuesday, September 30, 2025 at 05:25:05 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+14 y^{\prime \prime }-20 y^{\prime }+25 y&={\mathrm e}^{x} \left (\left (2+6 x \right ) \cos \left (2 x \right )+3 \sin \left (2 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 46

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-4*diff(diff(diff(y(x),x),x),x)+14*diff(diff(y(x),x),x)-20*diff(y(x),x)+25*y(x) = exp(x)*((2+6*x)*cos(2*x)+3*sin(2*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\left (x^{3}+x^{2}+\left (-16 c_3 +\frac {3}{8}\right ) x -16 c_1 -\frac {3}{8}\right ) \cos \left (2 x \right )-16 \sin \left (2 x \right ) \left (\left (c_4 +\frac {1}{16}\right ) x +c_2 +\frac {3}{256}\right )\right ) {\mathrm e}^{x}}{16} \]

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 62

ode=D[y[x],{x,4}]-4*D[y[x],{x,3}]+14*D[y[x],{x,2}]-20*D[y[x],x]+25*y[x]==Exp[x]*((2+6*x)*Cos[2*x]+3*Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{256} e^x \left (\left (-16 x^3-16 x^2+(6+256 c_4) x+6+256 c_3\right ) \cos (2 x)+(8 (3+32 c_2) x+3+256 c_1) \sin (2 x)\right ) \end{align*}

✓ Sympy. Time used: 0.544 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(((-6*x - 2)*cos(2*x) - 3*sin(2*x))*exp(x) + 25*y(x) - 20*Derivative(y(x), x) + 14*Derivative(y(x), (x, 2)) - 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\left (C_{1} + C_{2} x\right ) \sin {\left (2 x \right )} + \left (C_{3} + x \left (C_{4} - \frac {x^{2}}{16} - \frac {x}{16}\right )\right ) \cos {\left (2 x \right )}\right ) e^{x} \]