ODE
\[ y''(x) \left (a \left (x y'(x)-y(x)\right )+y'(x)^2\right )=b \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.460288 (sec), leaf count = 265
\[\left \{\text {Solve}\left [x=\int \frac {\frac {a^2 x^2}{4}+a y(x)+\sqrt {a b x^2-2 c_1+4 b y(x)}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2+2 c_1-b \left (a x^2+4 y(x)\right )\right ) \left (\frac {a^2 x^2}{4}+a y(x)+\sqrt {a b x^2-2 c_1+4 b y(x)}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )+c_2,y(x)\right ],\text {Solve}\left [x+\int \frac {\frac {a^2 x^2}{4}+a y(x)+\sqrt {a b x^2-2 c_1+4 b y(x)}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2+2 c_1-b \left (a x^2+4 y(x)\right )\right ) \left (\frac {a^2 x^2}{4}+a y(x)+\sqrt {a b x^2-2 c_1+4 b y(x)}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )=c_2,y(x)\right ]\right \}\]
Maple ✓
cpu = 1.945 (sec), leaf count = 423
\[\left [y \left (x \right ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_C1} \textit {\_f} a +\sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_f}^{2} a^{2}-4 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, b \textit {\_f} +2 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_C1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 \textit {\_C1}}d \textit {\_f} +\textit {\_C2} \right ), y \left (x \right ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_C1} \textit {\_f} a -\sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_f}^{2} a^{2}+4 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, b \textit {\_f} -2 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_C1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 \textit {\_C1}}d \textit {\_f} +\textit {\_C2} \right ), y \left (x \right ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x -\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_C1} \textit {\_f} a +\sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_f}^{2} a^{2}-4 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, b \textit {\_f} +2 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_C1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 \textit {\_C1}}d \textit {\_f} \right )+\textit {\_C2} \right ), y \left (x \right ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x -\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_C1} \textit {\_f} a -\sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_f}^{2} a^{2}+4 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, b \textit {\_f} -2 \sqrt {4 \textit {\_f} b -2 \textit {\_C1}}\, \textit {\_C1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 \textit {\_C1}}d \textit {\_f} \right )+\textit {\_C2} \right )\right ]\] Mathematica raw input
DSolve[(y'[x]^2 + a*(-y[x] + x*y'[x]))*y''[x] == b,y[x],x]
Mathematica raw output
{Solve[x == C[2] + Inactive[Integrate][((a^2*x^2)/4 + a*y[x] + Sqrt[a*b*x^2 - 2*C[1] + 4*b*y[x]])/Sqrt[(2*C[1] + a^2*((a*x^2)/4 + y[x])^2 - b*(a*x^2 + 4*y[x]))*((a^2*x^2)/4 + a*y[x] + Sqrt[a*b*x^2 - 2*C[1] + 4*b*y[x]])], (a*x^2)/4 + y[x]], y[x]], Solve[x + Inactive[Integrate][((a^2*x^2)/4 + a*y[x] + Sqrt[a*b*x^2 - 2*C[1] + 4*b*y[x]])/Sqrt[(2*C[1] + a^2*((a*x^2)/4 + y[x])^2 - b*(a*x^2 + 4*y[x]))*((a^2*x^2)/4 + a*y[x] + Sqrt[a*b*x^2 - 2*C[1] + 4*b*y[x]])], (a*x^2)/4 + y[x]] == C[2], y[x]]}
Maple raw input
dsolve((diff(y(x),x)^2+a*(x*diff(y(x),x)-y(x)))*diff(diff(y(x),x),x) = b, y(x))
Maple raw output
[y(x) = -1/4*a*x^2+RootOf(-x+Intat(1/(_f^2*a^2-4*_f*b+2*_C1)*(_f^3*a^3-4*_f^2*a*b+2*_C1*_f*a+(4*_f*b-2*_C1)^(1/2)*_f^2*a^2-4*(4*_f*b-2*_C1)^(1/2)*b*_f+2*(4*_f*b-2*_C1)^(1/2)*_C1)^(1/2),_f = _Z)+_C2), y(x) = -1/4*a*x^2+RootOf(-x+Intat(1/(_f^2*a^2-4*_f*b+2*_C1)*(_f^3*a^3-4*_f^2*a*b+2*_C1*_f*a-(4*_f*b-2*_C1)^(1/2)*_f^2*a^2+4*(4*_f*b-2*_C1)^(1/2)*b*_f-2*(4*_f*b-2*_C1)^(1/2)*_C1)^(1/2),_f = _Z)+_C2), y(x) = -1/4*a*x^2+RootOf(-x-Intat(1/(_f^2*a^2-4*_f*b+2*_C1)*(_f^3*a^3-4*_f^2*a*b+2*_C1*_f*a+(4*_f*b-2*_C1)^(1/2)*_f^2*a^2-4*(4*_f*b-2*_C1)^(1/2)*b*_f+2*(4*_f*b-2*_C1)^(1/2)*_C1)^(1/2),_f = _Z)+_C2), y(x) = -1/4*a*x^2+RootOf(-x-Intat(1/(_f^2*a^2-4*_f*b+2*_C1)*(_f^3*a^3-4*_f^2*a*b+2*_C1*_f*a-(4*_f*b-2*_C1)^(1/2)*_f^2*a^2+4*(4*_f*b-2*_C1)^(1/2)*b*_f-2*(4*_f*b-2*_C1)^(1/2)*_C1)^(1/2),_f = _Z)+_C2)]