64.11.49 problem 49

Internal problem ID [13420]
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 : Wednesday, March 05, 2025 at 09:56:43 PM
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: 0.245 (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: 0.281 (sec). Leaf size: 202

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]
 

\[ y(x)\to e^{2 x} \int _1^x\frac {1}{32} K[1]^2 \left (K[1]^2+e^{-2 K[1]} \sin (2 K[1])\right )dK[1]+e^{-2 x} \int _1^x-\frac {1}{32} e^{2 K[3]} K[3]^2 \left (e^{2 K[3]} K[3]^2+\sin (2 K[3])\right )dK[3]+\cos (2 x) \int _1^x\frac {1}{16} K[2]^2 \sin (2 K[2]) \left (e^{2 K[2]} K[2]^2+\sin (2 K[2])\right )dK[2]+\sin (2 x) \int _1^x-\frac {1}{16} \cos (2 K[4]) K[4]^2 \left (e^{2 K[4]} K[4]^2+\sin (2 K[4])\right )dK[4]+c_1 e^{2 x}+c_3 e^{-2 x}+c_2 \cos (2 x)+c_4 \sin (2 x) \]

✓ Sympy. Time used: 0.478 (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)
 

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