6.19.42 problem section 9.3, problem 42

Internal problem ID [2189]
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 42
Date solved : Tuesday, September 30, 2025 at 05:24:46 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime \prime }+13 y^{\prime \prime }-19 y^{\prime }+10 y&={\mathrm e}^{x} \left (\cos \left (2 x \right )+\sin \left (2 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 43

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-5*diff(diff(diff(y(x),x),x),x)+13*diff(diff(y(x),x),x)-19*diff(y(x),x)+10*y(x) = exp(x)*(cos(2*x)+sin(2*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{x} \left (\frac {\cos \left (2 x \right ) \left (60 x +800 c_3 -11\right )}{20}-\left (x -40 c_4 +\frac {73}{20}\right ) \sin \left (2 x \right )+40 c_2 \,{\mathrm e}^{x}+40 c_1 \right )}{40} \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 53

ode=1*D[y[x],{x,4}]-5*D[y[x],{x,3}]+13*D[y[x],{x,2}]-19*D[y[x],x]+10*y[x]==Exp[x]*(Cos[2*x]+Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{400} e^x \left (400 \left (c_4 e^x+c_3\right )+(30 x-13+400 c_2) \cos (2 x)-2 (5 x+17-200 c_1) \sin (2 x)\right ) \end{align*}

✓ Sympy. Time used: 0.385 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-sin(2*x) - cos(2*x))*exp(x) + 10*y(x) - 19*Derivative(y(x), x) + 13*Derivative(y(x), (x, 2)) - 5*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} e^{x} + C_{3} \sin {\left (2 x \right )} + C_{4} \cos {\left (2 x \right )} + \frac {\sqrt {2} x \sin {\left (2 x + \frac {\pi }{4} \right )}}{40} + \frac {\sqrt {2} x \cos {\left (2 x + \frac {\pi }{4} \right )}}{20}\right ) e^{x} \]