15.14.13 problem 13

Internal problem ID [3185]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 23, page 106
Problem number : 13
Date solved : Sunday, March 30, 2025 at 01:20:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 61

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

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

✓ Mathematica. Time used: 0.211 (sec). Leaf size: 64

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

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

✓ Sympy. Time used: 0.332 (sec). Leaf size: 88

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) - sin(k*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {k^{2} \sin {\left (k x \right )}}{k^{4} + 2 k^{2} n^{4} + n^{8}} - \frac {2 k n^{2} \cos {\left (k x \right )}}{k^{4} + 2 k^{2} n^{4} + n^{8}} + \frac {n^{4} \sin {\left (k x \right )}}{k^{4} + 2 k^{2} n^{4} + n^{8}} + \left (C_{1} + C_{2} x\right ) e^{- n^{2} x} \]