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 = 0.376121 (sec), leaf count = 102

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

Maple ✓
cpu = 0.584 (sec), leaf count = 58

\[\left [\int _{}^{y \left (x \right )}\frac {1}{a \,\textit {\_a}^{2}+\RootOf \left (\left (-a \right )^{\frac {1}{3}} \AiryBi \left (\textit {\_Z} \right ) \textit {\_C1} \textit {\_a} +\AiryAi \left (\textit {\_Z} \right ) \left (-a \right )^{\frac {1}{3}} \textit {\_a} +\AiryBi \left (1, \textit {\_Z}\right ) \textit {\_C1} +\AiryAi \left (1, \textit {\_Z}\right )\right ) \left (-a \right )^{\frac {1}{3}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] 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))

Maple raw output

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