60.7.218 problem 1809 (book 6.218)

Internal problem ID [11807]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1809 (book 6.218)
Date solved : Monday, January 27, 2025 at 11:36:48 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✗ Solution by Maple

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

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 21.257 (sec). Leaf size: 306

DSolve[y[x]*(1 + a^2 - 2*a^2*y[x]^2)*D[y[x],x]^2 + b*Sqrt[(1 - y[x]^2)*(1 - a^2*y[x]^2)]*D[y[x],x]^2 + (-1 + y[x]^2)*(-1 + a^2*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 {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 a^2 K[1]^3-a^2 K[1]-K[1]-b \sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 a^2 K[1]^3-a^2 K[1]-K[1]-b \sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 a^2 K[1]^3-a^2 K[1]-K[1]-b \sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ \end{align*}