4.17.21 \(a y(x) y'(x)-a x+y'(x)^2=0\)

ODE
\[ a y(x) y'(x)-a x+y'(x)^2=0 \] ODE Classification

[_dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 1.46942 (sec), leaf count = 45

\[\text {Solve}\left [\left \{x=\frac {\text {K$\$$230729} \left (a c_1+\sin ^{-1}(\text {K$\$$230729})\right )}{a \sqrt {1-\text {K$\$$230729}^2}},\frac {\text {K$\$$230729}}{a}+y(x)=\frac {x}{\text {K$\$$230729}}\right \},\{y(x),\text {K$\$$230729}\}\right ]\]

Maple
cpu = 0.041 (sec), leaf count = 106

\[ \left \{ [x \left ({\it \_T} \right ) ={{\it \_T} \left (-{\frac {1}{a}\sqrt {{\it \_T}-1}\sqrt {{\it \_T}+1}\ln \left ({\it \_T}+\sqrt {{{\it \_T}}^{2}-1} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}-1}}}}+{\it \_C1} \right ) {\frac {1}{\sqrt {{\it \_T}-1}}}{\frac {1}{\sqrt {{\it \_T}+1}}}},y \left ({\it \_T} \right ) ={1 \left (-{\frac {1}{a}\sqrt {{\it \_T}-1}\sqrt {{\it \_T}+1}\ln \left ({\it \_T}+\sqrt {{{\it \_T}}^{2}-1} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}-1}}}}+{\it \_C1} \right ) {\frac {1}{\sqrt {{\it \_T}-1}}}{\frac {1}{\sqrt {{\it \_T}+1}}}}-{\frac {{\it \_T}}{a}}] \right \} \] Mathematica raw input

DSolve[-(a*x) + a*y[x]*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

Solve[{x == (K$230729*(ArcSin[K$230729] + a*C[1]))/(a*Sqrt[1 - K$230729^2]), K$2
30729/a + y[x] == x/K$230729}, {y[x], K$230729}]

Maple raw input

dsolve(diff(y(x),x)^2+a*y(x)*diff(y(x),x)-a*x = 0, y(x),'implicit')

Maple raw output

[x(_T) = 1/(_T-1)^(1/2)/(_T+1)^(1/2)*_T*(-1/a*(_T-1)^(1/2)*(_T+1)^(1/2)/(_T^2-1)
^(1/2)*ln(_T+(_T^2-1)^(1/2))+_C1), y(_T) = 1/(_T-1)^(1/2)/(_T+1)^(1/2)*(-1/a*(_T
-1)^(1/2)*(_T+1)^(1/2)/(_T^2-1)^(1/2)*ln(_T+(_T^2-1)^(1/2))+_C1)-_T/a]