64.12.15 problem 15

Internal problem ID [13448]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 15
Date solved : Monday, March 31, 2025 at 07:58:10 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.179 (sec). Leaf size: 79

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

\[ y(x)\to \cos (x) \int _1^x-\frac {\sin (K[1])}{\left (\cos \left (\frac {K[1]}{2}\right )+\sin \left (\frac {K[1]}{2}\right )\right )^2}dK[1]+\sin (x) \int _1^x\frac {\cos (K[2])}{\left (\cos \left (\frac {K[2]}{2}\right )+\sin \left (\frac {K[2]}{2}\right )\right )^2}dK[2]+c_1 \cos (x)+c_2 \sin (x) \]

✓ Sympy. Time used: 0.484 (sec). Leaf size: 49

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

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