60.2.19 problem Problem 19

Internal problem ID [15199]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 19
Date solved : Thursday, October 02, 2025 at 10:07:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=1-\frac {1}{\sin \left (x \right )} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+y(x) = 1-1/sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\sin \left (x \right ) \ln \left (\sin \left (x \right )\right )+\left (x +c_1 \right ) \cos \left (x \right )+\sin \left (x \right ) c_2 +1 \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to (x+c_1) \cos (x)+\sin (x) (-\log (\sin (x))+c_2)+1 \end{align*}

✓ Sympy. Time used: 0.559 (sec). Leaf size: 34

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

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