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 = 0.175888 (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 = 4.732 (sec), leaf count = 316
\[\left [y \left (x \right ) = 27 \textit {\_C1} \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {9 x^{2}-2 x}\right )^{-\frac {2 \sqrt {9}}{9}} \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {x \left (9 x -2\right )}\right )^{-\frac {5 \sqrt {9}}{18}} \sqrt {\frac {\frac {\left (-\frac {1}{2}+\frac {5 x}{2}\right ) \sqrt {16}}{2 \sqrt {x \left (9 x -2\right )}}+1}{\sqrt {\frac {-16 x^{2}+8 x -1}{x \left (9 x -2\right )}}}}\, x \sqrt {-1+4 x}\, {\mathrm e}^{2 \sqrt {9 x^{2}-2 x}-\frac {\sqrt {16}\, \sqrt {x \left (9 x -2\right )}}{2}}, y \left (x \right ) = \frac {\textit {\_C1} \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {9 x^{2}-2 x}\right )^{\frac {2 \sqrt {9}}{9}} \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {x \left (9 x -2\right )}\right )^{\frac {5 \sqrt {9}}{18}} x \sqrt {-1+4 x}\, {\mathrm e}^{-2 \sqrt {9 x^{2}-2 x}+\frac {\sqrt {16}\, \sqrt {x \left (9 x -2\right )}}{2}}}{27 \sqrt {\frac {\frac {\left (-\frac {1}{2}+\frac {5 x}{2}\right ) \sqrt {16}}{2 \sqrt {x \left (9 x -2\right )}}+1}{\sqrt {\frac {-16 x^{2}+8 x -1}{x \left (9 x -2\right )}}}}}, y \left (x \right ) = 0, y \left (x \right ) = \textit {\_C2} \,x^{3}+\textit {\_C1} x +\frac {\textit {\_C1}^{2}}{\textit {\_C2}}\right ]\] 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))
Maple raw output
[y(x) = 27*_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*(-1+4*x)^(1/2)*exp(2*(9*x^2-2*x)^(1/2)-1/2*16^(1/2)*(x*(9*x-2))^(1/2)), y(x) = 1/27*_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*(-1+4*x)^(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]