29.15.2 problem 410

Internal problem ID [5008]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 15
Problem number : 410
Date solved : Monday, January 27, 2025 at 10:02:46 AM
CAS classification : [_Bernoulli]

\begin{align*} 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 ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 78

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)
 

\[ y \left (x \right ) = \frac {\sec \left (\frac {x}{2}\right )^{2} \left (a +\cos \left (\frac {x}{2}\right )^{2}\right )^{\frac {1}{a}} \cos \left (\frac {x}{2}\right )^{-\frac {2}{a}}}{\int \tan \left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )^{2} \cos \left (\frac {x}{2}\right )^{-\frac {2}{a}} \left (a +\cos \left (\frac {x}{2}\right )^{2}\right )^{\frac {-a +1}{a}}d x +c_{1}} \]

✓ Solution by Mathematica

Time used: 1.577 (sec). Leaf size: 74

DSolve[D[y[x],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*}