Internal problem ID [3156]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 15
Problem number: 410.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime } \left (a +\cos ^{2}\left (\frac {x}{2}\right )\right )-y \tan \left (\frac {x}{2}\right ) \left (1+a +\cos ^{2}\left (\frac {x}{2}\right )-y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 125
dsolve(diff(y(x),x)*(a+cos(1/2*x)^2) = y(x)*tan(1/2*x)*(1+a+cos(1/2*x)^2-y(x)),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (2 a +\cos \relax (x )+1\right )^{\frac {1}{a}} \left (1+\cos \relax (x )\right )^{-\frac {1}{a}}}{\cos \relax (x ) \left (\int \frac {2 \left (2 a +\cos \relax (x )+1\right )^{\frac {1}{a}} \left (1+\cos \relax (x )\right )^{-\frac {1}{a}} \tan \left (\frac {x}{2}\right )}{\left (1+\cos \relax (x )\right ) \left (2 a +\cos \relax (x )+1\right )}d x \right )+\cos \relax (x ) c_{1}+\int \frac {2 \left (2 a +\cos \relax (x )+1\right )^{\frac {1}{a}} \left (1+\cos \relax (x )\right )^{-\frac {1}{a}} \tan \left (\frac {x}{2}\right )}{\left (1+\cos \relax (x )\right ) \left (2 a +\cos \relax (x )+1\right )}d x +c_{1}} \]
✓ Solution by Mathematica
Time used: 1.345 (sec). Leaf size: 60
DSolve[y'[x](a+Cos[x/2]^2)==y[x] Tan[x/2](1+a+Cos[x/2]^2-y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{\frac {\sin ^2\left (\frac {x}{2}\right )}{a+1}+c_1 \left (a+\cos ^2\left (\frac {x}{2}\right )\right )^{-1/a} \cos ^{\frac {2}{a}+2}\left (\frac {x}{2}\right )} \\ y(x)\to 0 \\ \end{align*}