4.41.2 $\sqrt{a^2+x^2}(b{y}^{\prime }(x{)}^2+y(x)y^{″}(x))=y(x)y^{\prime }(x)$

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

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.214429 (sec), leaf count = 63

$$\left\{\{y(x)\to c_2((b+1)x\sqrt{a^2+x^2}+a^2(b+1)\log (\sqrt{a^2+x^2}+x)+(b+1)x^2+2c_1){}^{\frac{1}{b+1}}\}\right\}$$

Maple ✓
cpu = 0.087 (sec), leaf count = 52

$$\{\frac{y\left(x\right){\left(y\left(x\right)\right)}^b}{b+1}-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2}(a^2\ln (x+\sqrt{a^2+x^2})+x(x+\sqrt{a^2+x^2}))-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[Sqrt[a^2 + x^2]*(b*y'[x]^2 + y[x]*y''[x]) == y[x]*y'[x],y[x],x]

Mathematica raw output

{{y[x] -> C[2]*((1 + b)*x^2 + (1 + b)*x*Sqrt[a^2 + x^2] + 2*C[1] + a^2*(1 + b)*L
og[x + Sqrt[a^2 + x^2]])^(1 + b)^(-1)}}

Maple raw input

dsolve((a^2+x^2)^(1/2)*(y(x)*diff(diff(y(x),x),x)+b*diff(y(x),x)^2) = y(x)*diff(y(x),x), y(x),'implicit')

Maple raw output

y(x)/(b+1)*y(x)^b-1/2*(a^2*ln(x+(a^2+x^2)^(1/2))+x*(x+(a^2+x^2)^(1/2)))*_C1-_C2 
= 0