15.11.20 problem 20

Internal problem ID [3130]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 20
Date solved : Sunday, March 30, 2025 at 01:19:07 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 48

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*n^2*diff(diff(y(x),x),x)+n^4*y(x) = sin(k*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.53 (sec). Leaf size: 69

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

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

✓ Sympy. Time used: 0.211 (sec). Leaf size: 46

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

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