\[ a x y'(x)+b y(x)+c x^2+y'(x)^2=0 \] ✓ Mathematica : cpu = 5.01886 (sec), leaf count = 1118
DSolve[c*x^2 + b*y[x] + a*x*Derivative[1][y][x] + Derivative[1][y][x]^2 == 0,y[x],x]
\[\left \{\text {Solve}\left [\int _1^x\left (\frac {\sqrt {a^2 K[1]^2-4 c K[1]^2-4 b y(x)} y(x)}{c K[1]^4+2 a y(x) K[1]^2+b y(x) K[1]^2+4 y(x)^2}+\frac {c K[1]^3+a y(x) K[1]+b y(x) K[1]}{c K[1]^4+2 a y(x) K[1]^2+b y(x) K[1]^2+4 y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {\sqrt {a^2 x^2-4 c x^2-4 b K[2]} x}{2 \left (c x^4+2 a K[2] x^2+b K[2] x^2+4 K[2]^2\right )}-\int _1^x\left (\frac {a K[1]+b K[1]}{c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2}+\frac {\sqrt {a^2 K[1]^2-4 c K[1]^2-4 b K[2]}}{c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2}-\frac {2 b K[2]}{\sqrt {a^2 K[1]^2-4 c K[1]^2-4 b K[2]} \left (c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2\right )}-\frac {\left (2 a K[1]^2+b K[1]^2+8 K[2]\right ) \left (c K[1]^3+a K[2] K[1]+b K[2] K[1]\right )}{\left (c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2\right )^2}-\frac {K[2] \left (2 a K[1]^2+b K[1]^2+8 K[2]\right ) \sqrt {a^2 K[1]^2-4 c K[1]^2-4 b K[2]}}{\left (c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2\right )^2}\right )dK[1]+\frac {a x^2+4 K[2]}{2 \left (c x^4+2 a K[2] x^2+b K[2] x^2+4 K[2]^2\right )}\right )dK[2]=c_1,y(x)\right ],\text {Solve}\left [\int _1^x\left (\frac {c K[3]^3+a y(x) K[3]+b y(x) K[3]}{c K[3]^4+2 a y(x) K[3]^2+b y(x) K[3]^2+4 y(x)^2}-\frac {y(x) \sqrt {a^2 K[3]^2-4 c K[3]^2-4 b y(x)}}{c K[3]^4+2 a y(x) K[3]^2+b y(x) K[3]^2+4 y(x)^2}\right )dK[3]+\int _1^{y(x)}\left (\frac {\sqrt {a^2 x^2-4 c x^2-4 b K[4]} x}{2 \left (c x^4+2 a K[4] x^2+b K[4] x^2+4 K[4]^2\right )}-\int _1^x\left (\frac {a K[3]+b K[3]}{c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2}-\frac {\sqrt {a^2 K[3]^2-4 c K[3]^2-4 b K[4]}}{c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2}+\frac {2 b K[4]}{\sqrt {a^2 K[3]^2-4 c K[3]^2-4 b K[4]} \left (c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2\right )}-\frac {\left (2 a K[3]^2+b K[3]^2+8 K[4]\right ) \left (c K[3]^3+a K[4] K[3]+b K[4] K[3]\right )}{\left (c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2\right )^2}+\frac {K[4] \left (2 a K[3]^2+b K[3]^2+8 K[4]\right ) \sqrt {a^2 K[3]^2-4 c K[3]^2-4 b K[4]}}{\left (c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2\right )^2}\right )dK[3]+\frac {a x^2+4 K[4]}{2 \left (c x^4+2 a K[4] x^2+b K[4] x^2+4 K[4]^2\right )}\right )dK[4]=c_1,y(x)\right ]\right \}\] ✗ Maple : cpu = 0. (sec), leaf count = 0
dsolve(diff(y(x),x)^2+a*x*diff(y(x),x)+b*y(x)+c*x^2 = 0,y(x))
, exception
time expired