4.41.2 \(\sqrt {a^2+x^2} \left (b y'(x)^2+y(x) y''(x)\right )=y(x) y'(x)\)

ODE
\[ \sqrt {a^2+x^2} \left (b y'(x)^2+y(x) y''(x)\right )=y(x) y'(x) \] 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

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

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

\[ \left \{ {\frac {y \left ( x \right ) \left ( y \left ( x \right ) \right ) ^{b}}{b+1}}-{\frac {{\it \_C1}}{2} \left ( {a}^{2}\ln \left ( x+\sqrt {{a}^{2}+{x}^{2}} \right ) +x \left ( x+\sqrt {{a}^{2}+{x}^{2}} \right ) \right ) }-{\it \_C2}=0 \right \} \] 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