36.1.29 problem 27 part(c)

Internal problem ID [6284]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page 46
Problem number : 27 part(c)
Date solved : Monday, January 27, 2025 at 01:53:16 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\sqrt {1+\sin \left (x \right )}\, \left (1+y^{2}\right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.319 (sec). Leaf size: 37

dsolve([diff(y(x),x)=sqrt(1+sin(x))*(1+y(x)^2),y(0) = 1],y(x), singsol=all)
 

\[ y = \tan \left (\sqrt {2}\, \left (\int _{0}^{x}\operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\textit {\_z1}}{2}\right )\right ) \sin \left (\frac {\pi }{4}+\frac {\textit {\_z1}}{2}\right )d \textit {\_z1} \right )+\frac {\pi }{4}\right ) \]

✓ Solution by Mathematica

Time used: 0.394 (sec). Leaf size: 29

DSolve[{D[y[x],x]==Sqrt[1+Sin[x]]*(1+y[x]^2),{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \tan \left (\frac {1}{4} \left (8 \sin \left (\frac {x}{2}\right )-8 \cos \left (\frac {x}{2}\right )+\pi +8\right )\right ) \]