69.20.14 problem 653

Internal problem ID [18435]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 653
Date solved : Thursday, October 02, 2025 at 03:11:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 24

ode=D[y[x],{x,2}]+y[x]==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) \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 19

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

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