Internal problem ID [3664]
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]
\[ \boxed {y^{\prime } \left (a +\cos \left (\frac {x}{2}\right )^{2}\right )-y \tan \left (\frac {x}{2}\right ) \left (1+a +\cos \left (\frac {x}{2}\right )^{2}-y\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = \frac {\left (2 a +1+\cos \left (x \right )\right )^{\frac {1}{a}} \left (\cos \left (x \right )+1\right )^{-\frac {1}{a}}}{\cos \left (x \right ) \left (\int \frac {2 \left (2 a +1+\cos \left (x \right )\right )^{\frac {1}{a}} \left (\cos \left (x \right )+1\right )^{-\frac {1}{a}} \tan \left (\frac {x}{2}\right )}{\left (\cos \left (x \right )+1\right ) \left (2 a +1+\cos \left (x \right )\right )}d x \right )+\cos \left (x \right ) c_{1} +\int \frac {2 \left (2 a +1+\cos \left (x \right )\right )^{\frac {1}{a}} \left (\cos \left (x \right )+1\right )^{-\frac {1}{a}} \tan \left (\frac {x}{2}\right )}{\left (\cos \left (x \right )+1\right ) \left (2 a +1+\cos \left (x \right )\right )}d x +c_{1}} \]
✓ Solution by Mathematica
Time used: 1.657 (sec). Leaf size: 74
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 {(a+1) \left (a+\cos ^2\left (\frac {x}{2}\right )\right )^{\frac {1}{a}}}{\sin ^2\left (\frac {x}{2}\right ) \left (a+\cos ^2\left (\frac {x}{2}\right )\right )^{\frac {1}{a}}+(a+1) c_1 \cos ^{\frac {2}{a}+2}\left (\frac {x}{2}\right )} y(x)\to 0 \end{align*}