60.1.66 problem 66

Internal problem ID [10080]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 66
Date solved : Monday, January 27, 2025 at 06:24:09 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }-\frac {\sqrt {{| y \left (y-1\right ) \left (-1+a y\right )|}}}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 40

dsolve(diff(y(x),x) - sqrt(abs(y(x)*(1-y(x))*(1-a*y(x))))/sqrt(abs(x*(1-x)*(1-a*x)))=0,y(x), singsol=all)
 

\[ \int \frac {1}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}}d x -\int _{}^{y}\frac {1}{\sqrt {{| \textit {\_a} \left (\textit {\_a} -1\right ) \left (\textit {\_a} a -1\right )|}}}d \textit {\_a} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.454 (sec). Leaf size: 81

DSolve[D[y[x],x] - Sqrt[Abs[y[x]*(1-y[x])*(1-a*y[x])]]/Sqrt[Abs[x*(1-x)*(1-a*x)]]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {| (1-K[1]) K[1] (1-a K[1])| }}dK[1]\&\right ]\left [\int _1^x\frac {1}{\sqrt {| (K[2]-1) K[2] (a K[2]-1)| }}dK[2]+c_1\right ] \\ y(x)\to 0 \\ y(x)\to 1 \\ y(x)\to \frac {1}{a} \\ \end{align*}