60.7.203 problem 1794 (book 6.203)

Internal problem ID [11792]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1794 (book 6.203)
Date solved : Monday, January 27, 2025 at 11:35:55 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} a y \left (y-1\right ) y^{\prime \prime }-\left (a -1\right ) \left (2 y-1\right ) {y^{\prime }}^{2}+f y \left (y-1\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.163 (sec). Leaf size: 48

dsolve(a*y(x)*(-1+y(x))*diff(diff(y(x),x),x)-(a-1)*(2*y(x)-1)*diff(y(x),x)^2+f*y(x)*(-1+y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= 1 \\ y &= 0 \\ c_{1} {\mathrm e}^{-\frac {f x}{a}}-c_{2} +\int _{}^{y}\frac {\left (\textit {\_a} \left (\textit {\_a} -1\right )\right )^{\frac {1}{a}}}{\textit {\_a} \left (\textit {\_a} -1\right )}d \textit {\_a} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.224 (sec). Leaf size: 106

DSolve[f[x]*(-1 + y[x])*y[x]*D[y[x],x] - (-1 + a)*(-1 + 2*y[x])*D[y[x],x]^2 + a*(-1 + y[x])*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[1]}\left (-\frac {2}{a (K[1]-1)}+\frac {1}{a K[1] (K[1]-1)}-\frac {1}{K[1] (K[1]-1)}+\frac {2}{K[1]-1}\right )dK[1]\right )dK[1]\&\right ]\left [\int _1^x-\exp \left (-\int _1^{K[2]}\frac {f(K[2])}{a}dK[2]\right ) c_1dK[2]+c_2\right ] \]