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 = 1.42998 (sec), leaf count = 396
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (\frac {b F\left (\sin ^{-1}(K[2])|a^2\right ) \sqrt {1-K[2]^2} \sqrt {1-a^2 K[2]^2}}{\sqrt {\left (K[2]^2-1\right ) \left (a^2 K[2]^2-1\right )}}\right )}{\sqrt {c_1+2 \int _1^{K[2]}\frac {\exp \left (\frac {2 b F\left (\sin ^{-1}(K[1])|a^2\right ) \sqrt {1-K[1]^2} \sqrt {1-a^2 K[1]^2}}{\sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}\right ) \left (2 a^2 K[1]^3-a^2 K[1]-K[1]\right )}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {b F\left (\sin ^{-1}(K[3])|a^2\right ) \sqrt {1-K[3]^2} \sqrt {1-a^2 K[3]^2}}{\sqrt {\left (K[3]^2-1\right ) \left (a^2 K[3]^2-1\right )}}\right )}{\sqrt {c_1+2 \int _1^{K[3]}\frac {\exp \left (\frac {2 b F\left (\sin ^{-1}(K[1])|a^2\right ) \sqrt {1-K[1]^2} \sqrt {1-a^2 K[1]^2}}{\sqrt {\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}}\right ) \left (2 a^2 K[1]^3-a^2 K[1]-K[1]\right )}{\left (K[1]^2-1\right ) \left (a^2 K[1]^2-1\right )}dK[1]}}dK[3]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 9.091 (sec), leaf count = 550
\[\left [\int _{}^{y \left (x \right )}\frac {{\mathrm e}^{2 b \left (\int \frac {1}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g} \right )}}{\sqrt {-{\mathrm e}^{2 b \left (\int \frac {1}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g} \right )} \left (2 a^{2} \left (\int \frac {{\mathrm e}^{\int \frac {2 b}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g}} \textit {\_g}}{\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}d \textit {\_g} \right )-4 a^{2} \left (\int \frac {{\mathrm e}^{\int \frac {2 b}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g}} \textit {\_g}^{3}}{\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}d \textit {\_g} \right )+2 \left (\int \frac {{\mathrm e}^{\int \frac {2 b}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g}} \textit {\_g}}{\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}d \textit {\_g} \right )-\textit {\_C1} \right )}}d \textit {\_g} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {{\mathrm e}^{2 b \left (\int \frac {1}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g} \right )}}{\sqrt {-{\mathrm e}^{2 b \left (\int \frac {1}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g} \right )} \left (2 a^{2} \left (\int \frac {{\mathrm e}^{\int \frac {2 b}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g}} \textit {\_g}}{\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}d \textit {\_g} \right )-4 a^{2} \left (\int \frac {{\mathrm e}^{\int \frac {2 b}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g}} \textit {\_g}^{3}}{\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}d \textit {\_g} \right )+2 \left (\int \frac {{\mathrm e}^{\int \frac {2 b}{\sqrt {\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}}d \textit {\_g}} \textit {\_g}}{\textit {\_g}^{4} a^{2}-\textit {\_g}^{2} a^{2}-\textit {\_g}^{2}+1}d \textit {\_g} \right )-\textit {\_C1} \right )}}d \textit {\_g} -x -\textit {\_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
{{y[x] -> InverseFunction[Inactive[Integrate][-(E^((b*EllipticF[ArcSin[K[2]], a^2]*Sqrt[1 - K[2]^2]*Sqrt[1 - a^2*K[2]^2])/Sqrt[(-1 + K[2]^2)*(-1 + a^2*K[2]^2)])/Sqrt[C[1] + 2*Inactive[Integrate][(E^((2*b*EllipticF[ArcSin[K[1]], a^2]*Sqrt[1 - K[1]^2]*Sqrt[1 - a^2*K[1]^2])/Sqrt[(-1 + K[1]^2)*(-1 + a^2*K[1]^2)])*(-K[1] - a^2*K[1] + 2*a^2*K[1]^3))/((-1 + K[1]^2)*(-1 + a^2*K[1]^2)), {K[1], 1, K[2]}]]), {K[2], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][E^((b*EllipticF[ArcSin[K[3]], a^2]*Sqrt[1 - K[3]^2]*Sqrt[1 - a^2*K[3]^2])/Sqrt[(-1 + K[3]^2)*(-1 + a^2*K[3]^2)])/Sqrt[C[1] + 2*Inactive[Integrate][(E^((2*b*EllipticF[ArcSin[K[1]], a^2]*Sqrt[1 - K[1]^2]*Sqrt[1 - a^2*K[1]^2])/Sqrt[(-1 + K[1]^2)*(-1 + a^2*K[1]^2)])*(-K[1] - a^2*K[1] + 2*a^2*K[1]^3))/((-1 + K[1]^2)*(-1 + a^2*K[1]^2)), {K[1], 1, K[3]}]], {K[3], 1, #1}] & ][x + C[2]]}}
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))
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))*(2*a^2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2/(_g^4*a^2-_g^2*a^2-_g^2+1)*_g,_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^4*a^2-_g^2*a^2-_g^2+1)*_g^3,_g)+2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2/(_g^4*a^2-_g^2*a^2-_g^2+1)*_g,_g)-_C1))^(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))*(2*a^2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2/(_g^4*a^2-_g^2*a^2-_g^2+1)*_g,_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^4*a^2-_g^2*a^2-_g^2+1)*_g^3,_g)+2*Int(exp(b*Int(1/(_g^4*a^2-_g^2*a^2-_g^2+1)^(1/2),_g))^2/(_g^4*a^2-_g^2*a^2-_g^2+1)*_g,_g)-_C1))^(1/2),_g = y(x))-x-_C2 = 0]