4.19.29 \(a x^2 y'(x)^2+(1-a) a x^2-2 a x y(x) y'(x)+y(x)^2=0\)

ODE
\[ a x^2 y'(x)^2+(1-a) a x^2-2 a x y(x) y'(x)+y(x)^2=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous ODE, \(x^n f\left (\frac {y}{x} , y' \right )=0\), Solve for \(p\)

Mathematica
cpu = 0.143396 (sec), leaf count = 113

\[\left \{\left \{y(x)\to \frac {1}{2} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (e^{2 c_1}-a x^{2 \sqrt {\frac {a-1}{a}}}\right )\right \},\left \{y(x)\to \frac {1}{2} e^{c_1} x^{\sqrt {\frac {a-1}{a}}+1}-\frac {1}{2} a e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}}\right \}\right \}\]

Maple
cpu = 0.103 (sec), leaf count = 99

\[ \left \{ a{x}^{2}+ \left (y \relax (x ) \right ) ^{2}=0,\ln \relax (x ) -\int ^{{\frac {y \relax (x ) }{x}}}\!{\frac {1}{ \left (a-1 \right ) \left ({{\it \_a}}^{2}+a \right ) }\sqrt { \left (a-1 \right ) \left ({{\it \_a}}^{2}+a \right ) a}}{d{\it \_a}}-{\it \_C1}=0,\ln \relax (x ) +\int ^{{\frac {y \relax (x ) }{x}}}\!{\frac {1}{ \left (a-1 \right ) \left ({{\it \_a}}^{2}+a \right ) }\sqrt { \left (a-1 \right ) \left ({{\it \_a}}^{2}+a \right ) a}}{d{\it \_a}}-{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(1 - a)*a*x^2 + y[x]^2 - 2*a*x*y[x]*y'[x] + a*x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x^(1 - Sqrt[(-1 + a)/a])*(E^(2*C[1]) - a*x^(2*Sqrt[(-1 + a)/a])))/(2*
E^C[1])}, {y[x] -> -(a*x^(1 - Sqrt[(-1 + a)/a]))/(2*E^C[1]) + (E^C[1]*x^(1 + Sqr
t[(-1 + a)/a]))/2}}

Maple raw input

dsolve(a*x^2*diff(y(x),x)^2-2*a*x*y(x)*diff(y(x),x)+a*(1-a)*x^2+y(x)^2 = 0, y(x),'implicit')

Maple raw output

a*x^2+y(x)^2 = 0, ln(x)-Intat(((a-1)*(_a^2+a)*a)^(1/2)/(a-1)/(_a^2+a),_a = y(x)/
x)-_C1 = 0, ln(x)+Intat(((a-1)*(_a^2+a)*a)^(1/2)/(a-1)/(_a^2+a),_a = y(x)/x)-_C1
 = 0