83.17.2 problem 1 (b)

Internal problem ID [22048]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination II. page 256
Problem number : 1 (b)
Date solved : Thursday, October 02, 2025 at 08:23:10 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} k^{2} y^{\prime \prime }+2 k y^{\prime }+\left (k^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

ode:=k^2*diff(diff(y(x),x),x)+2*k*diff(y(x),x)+(k^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 31

ode=k^2*D[y[x],{x,2}]+2*k*D[y[x],x]+(k^2+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (c_2 e^{2 i x}+c_1\right ) e^{-\left (\left (\frac {1}{k}+i\right ) x\right )} \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(k**2*Derivative(y(x), (x, 2)) + 2*k*Derivative(y(x), x) + (k**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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