ODE
\[ b \sqrt {\left (1-y(x)^2\right ) \left (1-a^2 y(x)^2\right )} y'(x)^2+\left (1-y(x)^2\right ) \left (1-a^2 y(x)^2\right ) y''(x)+y(x) \left (-2 a^2 y(x)^2+a^2+1\right )=0 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✗
cpu = 600.002 (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 5.438 (sec), leaf count = 532
\[ \left \{ \int ^{y \left ( x \right ) }\!{1{{\rm e}^{2\,b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}}{\frac {1}{\sqrt {{{\rm e}^{2\,b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \left ( 4\,{a}^{2}\int \!{\frac { \left ( {{\rm e}^{b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \right ) ^{2}{{\it \_g}}^{3}}{1+{{\it \_g}}^{4}{a}^{2}+ \left ( -{a}^{2}-1 \right ) {{\it \_g}}^{2}}}\,{\rm d}{\it \_g}-2\,{a}^{2}\int \!{\frac { \left ( {{\rm e}^{b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \right ) ^{2}{\it \_g}}{1+{{\it \_g}}^{4}{a}^{2}+ \left ( -{a}^{2}-1 \right ) {{\it \_g}}^{2}}}\,{\rm d}{\it \_g}+{\it \_C1}-2\,\int \!{\frac { \left ( {{\rm e}^{b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \right ) ^{2}{\it \_g}}{1+{{\it \_g}}^{4}{a}^{2}+ \left ( -{a}^{2}-1 \right ) {{\it \_g}}^{2}}}\,{\rm d}{\it \_g} \right ) }}}}{d{\it \_g}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{1{{\rm e}^{2\,b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}}{\frac {1}{\sqrt {{{\rm e}^{2\,b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \left ( 4\,{a}^{2}\int \!{\frac { \left ( {{\rm e}^{b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \right ) ^{2}{{\it \_g}}^{3}}{1+{{\it \_g}}^{4}{a}^{2}+ \left ( -{a}^{2}-1 \right ) {{\it \_g}}^{2}}}\,{\rm d}{\it \_g}-2\,{a}^{2}\int \!{\frac { \left ( {{\rm e}^{b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \right ) ^{2}{\it \_g}}{1+{{\it \_g}}^{4}{a}^{2}+ \left ( -{a}^{2}-1 \right ) {{\it \_g}}^{2}}}\,{\rm d}{\it \_g}+{\it \_C1}-2\,\int \!{\frac { \left ( {{\rm e}^{b\int \!{\frac {1}{\sqrt {{{\it \_g}}^{4}{a}^{2}-{a}^{2}{{\it \_g}}^{2}-{{\it \_g}}^{2}+1}}}\,{\rm d}{\it \_g}}} \right ) ^{2}{\it \_g}}{1+{{\it \_g}}^{4}{a}^{2}+ \left ( -{a}^{2}-1 \right ) {{\it \_g}}^{2}}}\,{\rm d}{\it \_g} \right ) }}}}{d{\it \_g}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y[x]*(1 + a^2 - 2*a^2*y[x]^2) + b*Sqrt[(1 - y[x]^2)*(1 - a^2*y[x]^2)]*y'[x]^2 + (1 - y[x]^2)*(1 - a^2*y[x]^2)*y''[x] == 0,y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve((1-y(x)^2)*(1-a^2*y(x)^2)*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) = 0, y(x),'implicit')
Maple raw output
Intat(exp(2*b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))/(exp(2*b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))*(4*a^2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2*_g^3/(1+_g^4*a^2+(-a^2-1)*_g^2),_g)-2*a^2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2*_g/(1+_g^4*a^2+(-a^2-1)*_g^2),_g)+_C1-2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2*_g/(1+_g^4*a^2+(-a^2-1)*_g^2),_g)))^(1/2),_g = y(x))-x-_C2 = 0, Intat(-exp(2*b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))/(exp(2*b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))*(4*a^2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2*_g^3/(1+_g^4*a^2+(-a^2-1)*_g^2),_g)-2*a^2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2*_g/(1+_g^4*a^2+(-a^2-1)*_g^2),_g)+_C1-2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2*_g/(1+_g^4*a^2+(-a^2-1)*_g^2),_g)))^(1/2),_g = y(x))-x-_C2 = 0