87.12.35 problem 39 (a)

Internal problem ID [23483]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 93
Problem number : 39 (a)
Date solved : Thursday, October 02, 2025 at 09:42:16 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 \left (r +\beta \right ) y^{\prime }+r^{2} y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.065 (sec). Leaf size: 42

ode:=diff(diff(y(x),x),x)-2*(r+beta)*diff(y(x),x)+r^2*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sqrt {\beta \left (\beta +2 r \right )}\, {\mathrm e}^{x \left (r +\beta \right )} \sinh \left (x \sqrt {\beta \left (\beta +2 r \right )}\right )}{\beta \left (\beta +2 r \right )} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 61

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

\begin{align*} y(x)&\to \frac {e^{x \left (\beta +\sqrt {\beta (\beta +2 r)}+r\right )}-e^{x \left (\beta -\sqrt {\beta (\beta +2 r)}+r\right )}}{2 \sqrt {\beta (\beta +2 r)}} \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 63

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

\[ y{\left (x \right )} = - \frac {e^{x \left (b + r - \sqrt {b \left (b + 2 r\right )}\right )}}{2 \sqrt {b^{2} + 2 b r}} + \frac {e^{x \left (b + r + \sqrt {b \left (b + 2 r\right )}\right )}}{2 \sqrt {b^{2} + 2 b r}} \]