4.41.38 $b\sqrt{(1-y(x{)}^2)(1-a^{2}y(x{)}^2)}{y}^{\prime }(x{)}^2+(1-y(x{)}^2)(1-a^{2}y(x{)}^2)y^{″}(x)+y(x)(-2a^{2}y(x{)}^2+a^2+1)=0$

ODE
$$b\sqrt{(1-y(x{)}^2)(1-a^{2}y(x{)}^2)}{y}^{\prime }(x{)}^2+(1-y(x{)}^2)(1-a^{2}y(x{)}^2)y^{″}(x)+y(x)(-2a^{2}y(x{)}^2+a^2+1)=0$$ 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

$$\{{\int }^{y\left(x\right)}1{\mathrm{e}}^{2b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\frac{1}{\sqrt{{\mathrm{e}}^{2b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}(4a^2\int \frac{{\left({\mathrm{e}}^{b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\right)}^2{\mathit{\_}\mathit{g}}^3}{1+{\mathit{\_}\mathit{g}}^4{a}^2+(-a^2-1){\mathit{\_}\mathit{g}}^2}\mathrm{d}\mathit{\_}\mathit{g}-2a^2\int \frac{{\left({\mathrm{e}}^{b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\right)}^2\mathit{\_}\mathit{g}}{1+{\mathit{\_}\mathit{g}}^4{a}^2+(-a^2-1){\mathit{\_}\mathit{g}}^2}\mathrm{d}\mathit{\_}\mathit{g}+\mathit{\_}\mathit{C}\mathit{1}-2\int \frac{{\left({\mathrm{e}}^{b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\right)}^2\mathit{\_}\mathit{g}}{1+{\mathit{\_}\mathit{g}}^4{a}^2+(-a^2-1){\mathit{\_}\mathit{g}}^2}\mathrm{d}\mathit{\_}\mathit{g})}}d\mathit{\_}\mathit{g}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-1{\mathrm{e}}^{2b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\frac{1}{\sqrt{{\mathrm{e}}^{2b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}(4a^2\int \frac{{\left({\mathrm{e}}^{b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\right)}^2{\mathit{\_}\mathit{g}}^3}{1+{\mathit{\_}\mathit{g}}^4{a}^2+(-a^2-1){\mathit{\_}\mathit{g}}^2}\mathrm{d}\mathit{\_}\mathit{g}-2a^2\int \frac{{\left({\mathrm{e}}^{b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\right)}^2\mathit{\_}\mathit{g}}{1+{\mathit{\_}\mathit{g}}^4{a}^2+(-a^2-1){\mathit{\_}\mathit{g}}^2}\mathrm{d}\mathit{\_}\mathit{g}+\mathit{\_}\mathit{C}\mathit{1}-2\int \frac{{\left({\mathrm{e}}^{b\int \frac{1}{\sqrt{{\mathit{\_}\mathit{g}}^4{a}^2-a^2{\mathit{\_}\mathit{g}}^2-{\mathit{\_}\mathit{g}}^2+1}}\mathrm{d}\mathit{\_}\mathit{g}}\right)}^2\mathit{\_}\mathit{g}}{1+{\mathit{\_}\mathit{g}}^4{a}^2+(-a^2-1){\mathit{\_}\mathit{g}}^2}\mathrm{d}\mathit{\_}\mathit{g})}}d\mathit{\_}\mathit{g}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ 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