4.3.43 \(a x+2 y'(x)=\sqrt {a^2 x^2-4 b x^2-4 c y(x)}\)

ODE
\[ a x+2 y'(x)=\sqrt {a^2 x^2-4 b x^2-4 c y(x)} \] ODE Classification

[[_homogeneous, `class G`]]

Book solution method
Homogeneous equation, isobaric equation

Mathematica
cpu = 599.993 (sec), leaf count = 0 , timed out

$Aborted

Maple
cpu = 0.198 (sec), leaf count = 198

\[ \left \{ \int _{{\it \_b}}^{x}\!{1 \left (-a{\it \_a}+\sqrt {-4\,cy \relax (x ) + \left ({a}^{2}-4\,b \right ) {{\it \_a}}^{2}} \right ) \left (a{{\it \_a}}^{2}-{\it \_a}\,\sqrt {-4\,cy \relax (x ) + \left ({a}^{2}-4\,b \right ) {{\it \_a}}^{2}}+4\,y \relax (x ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \relax (x ) }\!2\, \left (-a{x}^{2}+x\sqrt { \left ({a}^{2}-4\,b \right ) {x}^{2}-4\,{\it \_f}\,c}-4\,{\it \_f} \right ) ^{-1}-\int _{{\it \_b}}^{x}\!{1 \left (-8\,{\frac {{\it \_f}\,c}{\sqrt { \left ({a}^{2}-4\,b \right ) {{\it \_a}}^{2}-4\,{\it \_f}\,c}}}+4\,a{\it \_a}-4\,\sqrt { \left ({a}^{2}-4\,b \right ) {{\it \_a}}^{2}-4\,{\it \_f}\,c} \right ) \left (a{{\it \_a}}^{2}-{\it \_a}\,\sqrt { \left ({a}^{2}-4\,b \right ) {{\it \_a}}^{2}-4\,{\it \_f}\,c}+4\,{\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[a*x + 2*y'[x] == Sqrt[a^2*x^2 - 4*b*x^2 - 4*c*y[x]],y[x],x]

Mathematica raw output

$Aborted

Maple raw input

dsolve(2*diff(y(x),x)+a*x = (a^2*x^2-4*b*x^2-4*c*y(x))^(1/2), y(x),'implicit')

Maple raw output

Int((-a*_a+(-4*c*y(x)+(a^2-4*b)*_a^2)^(1/2))/(a*_a^2-_a*(-4*c*y(x)+(a^2-4*b)*_a^
2)^(1/2)+4*y(x)),_a = _b .. x)+Intat(2/(-a*x^2+x*((a^2-4*b)*x^2-4*_f*c)^(1/2)-4*
_f)-Int((-8*c*_f/((a^2-4*b)*_a^2-4*_f*c)^(1/2)+4*a*_a-4*((a^2-4*b)*_a^2-4*_f*c)^
(1/2))/(a*_a^2-_a*((a^2-4*b)*_a^2-4*_f*c)^(1/2)+4*_f)^2,_a = _b .. x),_f = y(x))
+_C1 = 0