60.7.191 problem 1782 (book 6.191)

Internal problem ID [11780]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1782 (book 6.191)
Date solved : Monday, January 27, 2025 at 11:35:40 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.144 (sec). Leaf size: 19

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

\begin{align*} y &= -i \\ y &= i \\ y &= \tan \left (\ln \left (c_{1} x +c_{2} \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.409 (sec). Leaf size: 153

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 K[1]-1}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 K[1]-1}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 K[1]-1}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ \end{align*}