1.11.27 problem 27

Internal problem ID [348]
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 : 27
Date solved : Tuesday, September 30, 2025 at 03:57:38 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y&=\sin \left (x \right )+\cos \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 50

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+5*diff(diff(y(x),x),x)+4*y(x) = sin(x)+cos(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1}{9}+\frac {2 \left (-1+9 c_3 \right ) \cos \left (x \right )^{2}}{9}+\frac {\left (\left (-x +12 c_4 \right ) \sin \left (x \right )-x +6 c_1 \right ) \cos \left (x \right )}{6}+\frac {\left (18 c_2 +1\right ) \sin \left (x \right )}{18}-c_3 \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 50

ode=D[y[x],{x,4}]+5*D[y[x],{x,2}]+4*y[x]==Sin[x]+Cos[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (-\frac {11}{72}+c_1\right ) \cos (2 x)+\frac {1}{9} (-1+9 c_4) \sin (x)-\frac {1}{6} \cos (x) (x+(x-12 c_2) \sin (x)-6 c_3) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - sin(x) - cos(2*x) + 5*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} \sin {\left (x \right )} + C_{4} \cos {\left (2 x \right )} + \left (C_{1} - \frac {x}{6}\right ) \cos {\left (x \right )} + \left (C_{2} - \frac {x}{12}\right ) \sin {\left (2 x \right )} \]