7.11.29 problem 30

Internal problem ID [350]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 30
Date solved : Saturday, March 29, 2025 at 04:51:23 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y&=x^{2} \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 37

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-2*diff(diff(y(x),x),x)+y(x) = x^2*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_4 x +c_2 \right ) {\mathrm e}^{-x}+\frac {\left (x^{2}-4\right ) \cos \left (x \right )}{4}+\left (c_3 x +c_1 \right ) {\mathrm e}^{x}-\sin \left (x \right ) x \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 52

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

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

✓ Sympy. Time used: 0.221 (sec). Leaf size: 36

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

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