ODE
\[ (2-9 x) x^2 y''(x)^2+6 y(x) y''(x)-6 (1-6 x) x y'(x) y''(x)=36 x y'(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0301113 (sec), leaf count = 24
\[\left \{\left \{y(x)\to \frac {c_1^2 x^3}{c_2}+c_1 x+c_2\right \}\right \}\]
Maple ✓
cpu = 0.516 (sec), leaf count = 311
\[ \left \{ {{\it \_C1}\,x \left ( \left ( 9\,x-1 \right ) \sqrt {9}+9\,\sqrt {9\,{x}^{2}-2\,x} \right ) ^{{\frac {2\,\sqrt {9}}{9}}} \left ( \left ( 9\,x-1 \right ) \sqrt {9}+9\,\sqrt {x \left ( 9\,x-2 \right ) } \right ) ^{{\frac {5\,\sqrt {9}}{18}}}\sqrt {4\,x-1}{{\rm e}^{-2\,\sqrt {9\,{x}^{2}-2\,x}+{\frac {\sqrt {16}}{2}\sqrt {x \left ( 9\,x-2 \right ) }}}}{\frac {1}{\sqrt {{1 \left ( {\frac {\sqrt {16}}{2} \left ( -{\frac {1}{2}}+{\frac {5\,x}{2}} \right ) {\frac {1}{\sqrt {x \left ( 9\,x-2 \right ) }}}}+1 \right ) {\frac {1}{\sqrt {{\frac {-16\,{x}^{2}+8\,x-1}{x \left ( 9\,x-2 \right ) }}}}}}}}}}+y \left ( x \right ) =0,{\frac {{\it \_C1}\,\sqrt {5}\sqrt {4}x}{4} \left ( \left ( 9\,x-1 \right ) \sqrt {9}+9\,\sqrt {9\,{x}^{2}-2\,x} \right ) ^{-{\frac {2\,\sqrt {9}}{9}}} \left ( \left ( 9\,x-1 \right ) \sqrt {9}+9\,\sqrt {9\,{x}^{2}-2\,x} \right ) ^{-{\frac {5\,\sqrt {9}}{18}}}\sqrt {{1 \left ( {\frac {4}{5}}+{\sqrt {16} \left ( x-{\frac {1}{5}} \right ) {\frac {1}{\sqrt {9\,{x}^{2}-2\,x}}}} \right ) {\frac {1}{\sqrt {-{\frac { \left ( 4\,x-1 \right ) ^{2}}{9\,{x}^{2}-2\,x}}}}}}}\sqrt {4\,x-1}{{\rm e}^{-{\frac {-4+\sqrt {16}}{2}\sqrt {9\,{x}^{2}-2\,x}}}}}+y \left ( x \right ) =0,y \left ( x \right ) ={\it \_C2}\,{x}^{3}+{\it \_C1}\,x+{\frac {{{\it \_C1}}^{2}}{{\it \_C2}}} \right \} \] Mathematica raw input
DSolve[6*y[x]*y''[x] - 6*(1 - 6*x)*x*y'[x]*y''[x] + (2 - 9*x)*x^2*y''[x]^2 == 36*x*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] + (x^3*C[1]^2)/C[2] + C[2]}}
Maple raw input
dsolve(x^2*(2-9*x)*diff(diff(y(x),x),x)^2-6*x*(1-6*x)*diff(y(x),x)*diff(diff(y(x),x),x)+6*y(x)*diff(diff(y(x),x),x) = 36*x*diff(y(x),x)^2, y(x),'implicit')
Maple raw output
1/4*_C1*((9*x-1)*9^(1/2)+9*(9*x^2-2*x)^(1/2))^(-2/9*9^(1/2))*((9*x-1)*9^(1/2)+9*(9*x^2-2*x)^(1/2))^(-5/18*9^(1/2))*5^(1/2)*4^(1/2)*(1/(-(4*x-1)^2/(9*x^2-2*x))^(1/2)*(4/5+16^(1/2)*(x-1/5)/(9*x^2-2*x)^(1/2)))^(1/2)*x*(4*x-1)^(1/2)*exp(-1/2*(9*x^2-2*x)^(1/2)*(-4+16^(1/2)))+y(x) = 0, _C1*((9*x-1)*9^(1/2)+9*(9*x^2-2*x)^(1/2))^(2/9*9^(1/2))*((9*x-1)*9^(1/2)+9*(x*(9*x-2))^(1/2))^(5/18*9^(1/2))/((1/2*(-1/2+5/2*x)*16^(1/2)/(x*(9*x-2))^(1/2)+1)/((-16*x^2+8*x-1)/x/(9*x-2))^(1/2))^(1/2)*x*(4*x-1)^(1/2)*exp(-2*(9*x^2-2*x)^(1/2)+1/2*16^(1/2)*(x*(9*x-2))^(1/2))+y(x) = 0, y(x) = _C2*x^3+_C1*x+_C1^2/_C2