12.19.43 problem section 9.3, problem 43

Internal problem ID [2190]
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 43
Date solved : Tuesday, March 04, 2025 at 01:51:24 PM
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 }+39 y&={\mathrm e}^{-2 x} \left (\left (4-15 x \right ) \cos \left (3 x \right )-\left (4+15 x \right ) \sin \left (3 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.304 (sec). Leaf size: 57

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)+39*y(x) = exp(-2*x)*((4-15*x)*cos(3*x)-(4+15*x)*sin(3*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (-30 x^{2}+240 c_3 +30 x +11\right ) \cos \left (3 x \right )+30 \sin \left (3 x \right ) \left (x^{2}+x +8 c_4 -\frac {11}{30}\right )\right ) {\mathrm e}^{-2 x}}{240}+c_1 \,{\mathrm e}^{-3 x}+c_2 \,{\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.618 (sec). Leaf size: 73

ode=1*D[y[x],{x,4}]+8*D[y[x],{x,3}]+32*D[y[x],{x,2}]+64*D[y[x],x]+39*y[x]==Exp[-2*x]*((4-15*x)*Cos[3*x]-(4+15*x)*Sin[3*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{720} e^{-3 x} \left (-5 e^x \left (18 x^2-18 x-5-144 c_2\right ) \cos (3 x)+e^x \left (90 x^2+90 x-41+720 c_1\right ) \sin (3 x)+720 \left (c_4 e^{2 x}+c_3\right )\right ) \]

✓ Sympy. Time used: 1.429 (sec). Leaf size: 66

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-(4 - 15*x)*cos(3*x) + (15*x + 4)*sin(3*x))*exp(-2*x) + 39*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)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{4} e^{- 2 x} + \left (C_{2} \sin {\left (3 x \right )} + C_{3} \cos {\left (3 x \right )} - \frac {\sqrt {2} x^{2} \cos {\left (3 x + \frac {\pi }{4} \right )}}{8} + \frac {\sqrt {2} x \sin {\left (3 x + \frac {\pi }{4} \right )}}{8}\right ) e^{- x}\right ) e^{- x} \]