10.8.13 problem 19

Internal problem ID [1285]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.3 Complex Roots of the Characteristic Equation , page 164
Problem number : 19
Date solved : Tuesday, March 04, 2025 at 12:27:48 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=0\\ y^{\prime }\left (\frac {\pi }{2}\right )&=2 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+5*y(x) = 0; 
ic:=y(1/2*Pi) = 0, D(y)(1/2*Pi) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\sin \left (2 x \right ) {\mathrm e}^{-\frac {\pi }{2}+x} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 20

ode=D[y[x],{x,2}]-2*D[y[x],x]+5*y[x]==0; 
ic={y[Pi/2]==0,Derivative[1][y][Pi/2]==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -e^{x-\frac {\pi }{2}} \sin (2 x) \]

✓ Sympy. Time used: 0.168 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(pi/2): 0, Subs(Derivative(y(x), x), x, pi/2): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {e^{x} \sin {\left (2 x \right )}}{e^{\frac {\pi }{2}}} \]