ODE
\[ 9 y(x)^2 y'(x)^2-3 x y'(x)+y(x)=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _rational]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 0.148992 (sec), leaf count = 180
\[\left \{\left \{y(x)\to -\sqrt [3]{-2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )}\right \},\left \{y(x)\to \sqrt [3]{2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )}\right \},\left \{y(x)\to (-1)^{2/3} \sqrt [3]{2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )}\right \},\left \{y(x)\to \frac {\sqrt [3]{-e^{c_1} \left (e^{c_1}-2 x\right )}}{2^{2/3}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{-e^{c_1} \left (e^{c_1}-2 x\right )}}{2^{2/3}}\right \},\left \{y(x)\to \left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{-e^{c_1} \left (e^{c_1}-2 x\right )}\right \}\right \}\]
Maple ✓
cpu = 1.828 (sec), leaf count = 62
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{3}-{\frac {{x}^{2}}{4}}=0,\ln \left ( x \right ) -\int ^{{y \left ( x \right ) {x}^{-{\frac {2}{3}}}}}\!{\frac {1}{8\,{{\it \_a}}^{4}-2\,{\it \_a}} \left ( -12\,{{\it \_a}}^{3}+3\,\sqrt {-4\,{{\it \_a}}^{3}+1}+3 \right ) }{d{\it \_a}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y[x] - 3*x*y'[x] + 9*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((-2)^(1/3)*(E^C[1]*(-2*E^C[1] + x))^(1/3))}, {y[x] -> 2^(1/3)*(E^C[1]*(-2*E^C[1] + x))^(1/3)}, {y[x] -> (-1)^(2/3)*2^(1/3)*(E^C[1]*(-2*E^C[1] + x))^(1/3)}, {y[x] -> (-(E^C[1]*(E^C[1] - 2*x)))^(1/3)/2^(2/3)}, {y[x] -> -(((-1)^(1/3)*(-(E^C[1]*(E^C[1] - 2*x)))^(1/3))/2^(2/3))}, {y[x] -> (-1/2)^(2/3)*(-(E^C[1]*(E^C[1] - 2*x)))^(1/3)}}
Maple raw input
dsolve(9*y(x)^2*diff(y(x),x)^2-3*x*diff(y(x),x)+y(x) = 0, y(x),'implicit')
Maple raw output
y(x)^3-1/4*x^2 = 0, ln(x)-Intat((-12*_a^3+3*(-4*_a^3+1)^(1/2)+3)/(8*_a^4-2*_a),_a = y(x)/x^(2/3))-_C1 = 0