28.2.53 problem 53

Internal problem ID [4496]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 53
Date solved : Tuesday, March 04, 2025 at 06:49:05 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-18 y^{\prime \prime }+81 y&=72 \,{\mathrm e}^{3 x}+729 x^{2} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 45

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-18*diff(diff(y(x),x),x)+81*y(x) = 72*exp(3*x)+729*x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = 4+\frac {\left (1+6 x^{2}+2 \left (-2+3 c_4 \right ) x +6 c_{2} \right ) {\mathrm e}^{3 x}}{6}+\left (x c_3 +c_{1} \right ) {\mathrm e}^{-3 x}+9 x^{2} \]

✓ Mathematica. Time used: 0.307 (sec). Leaf size: 48

ode=D[y[x],{x,4}]-18*D[y[x],{x,2}]+81*y[x]==72*Exp[3*x]+729*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 9 x^2+e^{3 x} \left (x^2+\left (-\frac {2}{3}+c_4\right ) x+\frac {1}{6}+c_3\right )+e^{-3 x} (c_2 x+c_1)+4 \]

✓ Sympy. Time used: 0.181 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-729*x**2 + 81*y(x) - 72*exp(3*x) - 18*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 9 x^{2} + \left (C_{1} + C_{2} x\right ) e^{- 3 x} + \left (C_{3} + x \left (C_{4} + x\right )\right ) e^{3 x} + 4 \]