Internal problem ID [3655]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 401.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime } \sqrt {x \left (1-x \right ) \left (-x a +1\right )}-\sqrt {y \left (1-y\right ) \left (1-a y\right )}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
dsolve(diff(y(x),x)*sqrt(x*(1-x)*(-a*x+1)) = sqrt(y(x)*(1-y(x))*(1-a*y(x))),y(x), singsol=all)
\[ \int \frac {1}{\sqrt {x \left (x -1\right ) \left (a x -1\right )}}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (\textit {\_a} -1\right ) \left (a \textit {\_a} -1\right )}}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 17.58 (sec). Leaf size: 117
DSolve[y'[x] Sqrt[x (1-x)(1-a x)]==Sqrt[y[x](1-y[x])(1-a y[x])],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {ns}\left (\frac {1}{2} i \sqrt {a} c_1-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {x-1}}\right ),\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2 \left (-1+\text {sn}\left (\frac {1}{2} i \sqrt {a} c_1-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {x-1}}\right ),\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2\right ) y(x)\to 0 y(x)\to 1 y(x)\to \frac {1}{a} \end{align*}