4.37.3 \(y''(x)=a \left (2 y(x) y'(x)+1\right )\)

ODE
\[ y''(x)=a \left (2 y(x) y'(x)+1\right ) \] 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

\[\left \{\left \{y(x)\to \frac {a \left (c_2 \text {Ai}'\left (\frac {a \left (c_1-a x\right )}{\left (-a^2\right )^{2/3}}\right )+\text {Bi}'\left (\frac {a \left (c_1-a x\right )}{\left (-a^2\right )^{2/3}}\right )\right )}{\left (-a^2\right )^{2/3} \left (c_2 \text {Ai}\left (\frac {a \left (c_1-a x\right )}{\left (-a^2\right )^{2/3}}\right )+\text {Bi}\left (\frac {a \left (c_1-a x\right )}{\left (-a^2\right )^{2/3}}\right )\right )}\right \}\right \}\]

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

\[ \left \{ \int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{2}a+{\it RootOf} \left ( \sqrt [3]{-a}{{\rm Bi}\left ({\it \_Z}\right )}{\it \_C1}\,{\it \_a}+{{\rm Ai}\left ({\it \_Z}\right )}\sqrt [3]{-a}{\it \_a}+{{\rm Bi}^{(1)}\left ({\it \_Z}\right )}{\it \_C1}+{{\rm Ai}^{(1)}\left ({\it \_Z}\right )} \right ) \sqrt [3]{-a} \right ) ^{-1}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] 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