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

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 = 1.13249 (sec), leaf count = 43

\[\text {Solve}\left [\left \{x=\frac {\text {K$\$$230907} \left (a c_1+\sinh ^{-1}(\text {K$\$$230907})\right )}{a \sqrt {\text {K$\$$230907}^2+1}},\frac {x}{\text {K$\$$230907}}+y(x)=\frac {\text {K$\$$230907}}{a}\right \},\{y(x),\text {K$\$$230907}\}\right ]\]

Maple
cpu = 0.02 (sec), leaf count = 48

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

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

Mathematica raw output

Solve[{x == (K$230907*(ArcSinh[K$230907] + a*C[1]))/(a*Sqrt[1 + K$230907^2]), x/
K$230907 + y[x] == K$230907/a}, {y[x], K$230907}]

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