##### 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 = 3.1324 (sec), leaf count = 390

$\text {Solve}\left [c_1=\int _1^{y(x)}\frac {a x^2+\sqrt {a^2 x^2-4 b x^2-4 c K[2]} x+4 K[2]-2 \left (b x^4+K[2] \left (2 a x^2+c x^2+4 K[2]\right )\right ) \int _1^x\frac {\left (4 K[1]^2 K[2]^2-b K[1]^6\right ) a^2-K[1] \left (\sqrt {a^2 K[1]^2-4 b K[1]^2-4 c K[2]} \left (b K[1]^4+4 K[2]^2\right )-4 c K[1] K[2]^2\right ) a+2 \left (2 b K[1]^2+c K[2]\right ) \left (b K[1]^4+K[2] \left (c K[1]^2-2 \sqrt {a^2 K[1]^2-4 b K[1]^2-4 c K[2]} K[1]-4 K[2]\right )\right )}{\sqrt {a^2 K[1]^2-4 b K[1]^2-4 c K[2]} \left (b K[1]^4+K[2] \left (2 a K[1]^2+c K[1]^2+4 K[2]\right )\right )^2}dK[1]}{2 \left (b x^4+K[2] \left (2 a x^2+c x^2+4 K[2]\right )\right )}dK[2]+\int _1^x\frac {b K[1]^3+y(x) \left (a K[1]+c K[1]-\sqrt {\left (a^2-4 b\right ) K[1]^2-4 c y(x)}\right )}{b K[1]^4+(2 a+c) y(x) K[1]^2+4 y(x)^2}dK[1],y(x)\right ]$

Maple
cpu = 0.192 (sec), leaf count = 269

$\left [\int _{\textit {\_b}}^{x}-\frac {-a \textit {\_a} +\sqrt {\textit {\_a}^{2} a^{2}-4 \textit {\_a}^{2} b -4 c y \left (x \right )}}{-a \,\textit {\_a}^{2}+\textit {\_a} \sqrt {\textit {\_a}^{2} a^{2}-4 \textit {\_a}^{2} b -4 c y \left (x \right )}-4 y \left (x \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (\frac {2}{-a \,x^{2}+x \sqrt {a^{2} x^{2}-4 b \,x^{2}-4 \textit {\_f} c}-4 \textit {\_f}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {2 c}{\sqrt {\textit {\_a}^{2} a^{2}-4 \textit {\_a}^{2} b -4 \textit {\_f} c}\, \left (-a \,\textit {\_a}^{2}+\textit {\_a} \sqrt {\textit {\_a}^{2} a^{2}-4 \textit {\_a}^{2} b -4 \textit {\_f} c}-4 \textit {\_f} \right )}+\frac {\left (-a \textit {\_a} +\sqrt {\textit {\_a}^{2} a^{2}-4 \textit {\_a}^{2} b -4 \textit {\_f} c}\right ) \left (-\frac {2 \textit {\_a} c}{\sqrt {\textit {\_a}^{2} a^{2}-4 \textit {\_a}^{2} b -4 \textit {\_f} c}}-4\right )}{\left (-a \,\textit {\_a}^{2}+\textit {\_a} \sqrt {\textit {\_a}^{2} a^{2}-4 \textit {\_a}^{2} b -4 \textit {\_f} c}-4 \textit {\_f} \right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +\textit {\_C1} = 0\right ]$ 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

Solve[C[1] == Inactive[Integrate][(b*K[1]^3 + y[x]*(a*K[1] + c*K[1] - Sqrt[(a^2
- 4*b)*K[1]^2 - 4*c*y[x]]))/(b*K[1]^4 + (2*a + c)*K[1]^2*y[x] + 4*y[x]^2), {K[1]
, 1, x}] + Inactive[Integrate][(a*x^2 + 4*K[2] + x*Sqrt[a^2*x^2 - 4*b*x^2 - 4*c*
K[2]] - 2*(b*x^4 + K[2]*(2*a*x^2 + c*x^2 + 4*K[2]))*Inactive[Integrate][(a^2*(-(
b*K[1]^6) + 4*K[1]^2*K[2]^2) - a*K[1]*(-4*c*K[1]*K[2]^2 + Sqrt[a^2*K[1]^2 - 4*b*
K[1]^2 - 4*c*K[2]]*(b*K[1]^4 + 4*K[2]^2)) + 2*(2*b*K[1]^2 + c*K[2])*(b*K[1]^4 +
K[2]*(c*K[1]^2 - 4*K[2] - 2*K[1]*Sqrt[a^2*K[1]^2 - 4*b*K[1]^2 - 4*c*K[2]])))/(Sq
rt[a^2*K[1]^2 - 4*b*K[1]^2 - 4*c*K[2]]*(b*K[1]^4 + K[2]*(2*a*K[1]^2 + c*K[1]^2 +
 4*K[2]))^2), {K[1], 1, x}])/(2*(b*x^4 + K[2]*(2*a*x^2 + c*x^2 + 4*K[2]))), {K[2
], 1, y[x]}], y[x]]

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))

Maple raw output

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