87.12.34 problem 38 (a)

Internal problem ID [23482]
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 : 38 (a)
Date solved : Thursday, October 02, 2025 at 09:42:16 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 r y^{\prime }+\left (r^{2}-\frac {\alpha ^{2}}{4}\right ) 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: 0.022 (sec). Leaf size: 18

ode:=diff(diff(y(x),x),x)-2*r*diff(y(x),x)+(r^2-1/4*alpha^2)*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 28

ode=D[y[x],{x,2}]-2*r*D[y[x],x]+(r^2-\[Alpha]^2/4)*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 {\left (e^{\alpha x}-1\right ) e^{r x-\frac {\alpha x}{2}}}{\alpha } \end{align*}

✓ Sympy. Time used: 0.161 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
r = symbols("r") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*r*Derivative(y(x), x) + (-a**2/4 + r**2)*y(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)
 

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