87.18.18 problem 18

Internal problem ID [23659]
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 135
Problem number : 18
Date solved : Thursday, October 02, 2025 at 09:43:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\csc \left (x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 22

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

\[ y = -\ln \left (\csc \left (x \right )\right ) \sin \left (x \right )-\left (x -\frac {\pi }{2}+1\right ) \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to \frac {1}{2} (\pi -2 (x+1)) \cos (x)+\sin (x) \log (\sin (x)) \end{align*}

✓ Sympy. Time used: 0.209 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \left (- x - 1 + \frac {\pi }{2}\right ) \cos {\left (x \right )} + \log {\left (\sin {\left (x \right )} \right )} \sin {\left (x \right )} \]