28.2.52 problem 52

Internal problem ID [4495]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 52
Date solved : Tuesday, March 04, 2025 at 06:49:04 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+16 y&=32 \,{\mathrm e}^{2 x}+16 x^{3} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 43

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-8*diff(diff(y(x),x),x)+16*y(x) = 32*exp(2*x)+16*x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {\left (3+8 x^{2}+8 \left (c_4 -1\right ) x +8 c_{2} \right ) {\mathrm e}^{2 x}}{8}+\left (x c_3 +c_{1} \right ) {\mathrm e}^{-2 x}+x^{3}+3 x \]

✓ Mathematica. Time used: 0.315 (sec). Leaf size: 47

ode=D[y[x],{x,4}]-8*D[y[x],{x,2}]+16*y[x]==32*Exp[2*x]+16*x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x \left (x^2+3\right )+e^{2 x} \left (x^2+(-1+c_4) x+\frac {3}{8}+c_3\right )+e^{-2 x} (c_2 x+c_1) \]

✓ Sympy. Time used: 0.170 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*x**3 + 16*y(x) - 32*exp(2*x) - 8*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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