58.11.49 problem 49

Internal problem ID [14780]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 49
Date solved : Thursday, October 02, 2025 at 09:53:29 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-16 y&=x^{2} \sin \left (2 x \right )+x^{4} {\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 79

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-16*y(x) = x^2*sin(2*x)+x^4*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (128 x^{5}-480 x^{4}+800 x^{3}-600 x^{2}+20480 c_3 +60 x +105\right ) {\mathrm e}^{2 x}}{20480}+\frac {\left (8 x^{3}+768 c_1 -15 x \right ) \cos \left (2 x \right )}{768}+\frac {\left (-6 x^{2}+256 c_4 -11\right ) \sin \left (2 x \right )}{256}+c_2 \,{\mathrm e}^{-2 x} \]

✓ Mathematica. Time used: 0.334 (sec). Leaf size: 92

ode=D[y[x],{x,4}]-16*y[x]==x^2*Sin[2*x]+x^4*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{768} \left (8 x^3-15 x+768 c_2\right ) \cos (2 x)-\frac {1}{512} \left (24 x^2-5-1024 c_4\right ) \sin (x) \cos (x)+\frac {e^{2 x} \left (128 x^5-480 x^4+800 x^3-600 x^2+60 x+105+20480 c_1\right )}{20480}+c_3 e^{-2 x} \end{align*}

✓ Sympy. Time used: 0.304 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4*exp(2*x) - x**2*sin(2*x) - 16*y(x) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{4} e^{- 2 x} + \left (C_{1} - \frac {3 x^{2}}{128}\right ) \sin {\left (2 x \right )} + \left (C_{2} + \frac {x^{3}}{96} - \frac {5 x}{256}\right ) \cos {\left (2 x \right )} + \left (C_{3} + \frac {x^{5}}{160} - \frac {3 x^{4}}{128} + \frac {5 x^{3}}{128} - \frac {15 x^{2}}{512} + \frac {3 x}{1024}\right ) e^{2 x} \]