4.37.3 $y^{″}(x)=a(2y(x)y^{\prime }(x)+1)$

ODE
$$y^{″}(x)=a(2y(x)y^{\prime }(x)+1)$$ ODE Classification

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.276474 (sec), leaf count = 102

$$\left\{\{y(x)\to \frac{a(c_2{\text{Ai}}^{\prime }\left(\frac{a(c_1-ax)}{{(-a^2)}^{2/3}}\right)+{\text{Bi}}^{\prime }\left(\frac{a(c_1-ax)}{{(-a^2)}^{2/3}}\right))}{{(-a^2)}^{2/3}(c_2\text{Ai}\left(\frac{a(c_1-ax)}{{(-a^2)}^{2/3}}\right)+\text{Bi}\left(\frac{a(c_1-ax)}{{(-a^2)}^{2/3}}\right))}\}\right\}$$

Maple ✓
cpu = 0.189 (sec), leaf count = 58

$$\{{\int }^{y\left(x\right)}{({\mathit{\_}\mathit{a}}^{2}a+\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(\sqrt[3]{-a}\mathrm{B}\mathrm{i}\left(\mathit{\_}\mathit{Z}\right)\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{a}+\mathrm{A}\mathrm{i}\left(\mathit{\_}\mathit{Z}\right)\sqrt[3]{-a}\mathit{\_}\mathit{a}+{\mathrm{B}\mathrm{i}}^{(1)}\left(\mathit{\_}\mathit{Z}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathrm{A}\mathrm{i}}^{(1)}\left(\mathit{\_}\mathit{Z}\right))\sqrt[3]{-a})}^{-1}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[y''[x] == a*(1 + 2*y[x]*y'[x]),y[x],x]

Mathematica raw output

{{y[x] -> (a*(AiryBiPrime[(a*(-(a*x) + C[1]))/(-a^2)^(2/3)] + AiryAiPrime[(a*(-(
a*x) + C[1]))/(-a^2)^(2/3)]*C[2]))/((-a^2)^(2/3)*(AiryBi[(a*(-(a*x) + C[1]))/(-a
^2)^(2/3)] + AiryAi[(a*(-(a*x) + C[1]))/(-a^2)^(2/3)]*C[2]))}}

Maple raw input

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

Maple raw output

Intat(1/(_a^2*a+RootOf((-a)^(1/3)*AiryBi(_Z)*_C1*_a+AiryAi(_Z)*(-a)^(1/3)*_a+Air
yBi(1,_Z)*_C1+AiryAi(1,_Z))*(-a)^(1/3)),_a = y(x))-x-_C2 = 0