76.13.34 problem 34

Internal problem ID [17537]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 34
Date solved : Thursday, March 13, 2025 at 10:11:44 AM
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: 0.029 (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);
 

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

✓ Mathematica. Time used: 0.013 (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: 0.167 (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)
 

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