12.19.42 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, March 04, 2025 at 01:51:22 PM
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: 0.008 (sec). Leaf size: 48

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);
 

\[ y = \frac {3 \,{\mathrm e}^{x} \left (x +\frac {40 c_3}{3}+\frac {1}{15}\right ) \cos \left (2 x \right )}{40}-\frac {{\mathrm e}^{x} \left (x -40 c_4 +\frac {39}{10}\right ) \sin \left (2 x \right )}{40}+{\mathrm e}^{2 x} c_2 +\frac {{\mathrm e}^{x} \left (8 c_1 +1\right )}{8} \]

✓ Mathematica. Time used: 0.171 (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]
 

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

✓ Sympy. Time used: 0.602 (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)
 

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