Internal problem ID [7982]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 402.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {3 \left (y^{\prime }\right )^{2}+4 x y^{\prime }-y+x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.221 (sec). Leaf size: 111
dsolve(3*diff(y(x),x)^2+4*x*diff(y(x),x)-y(x)+x^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x^{2}}{3} \\ y \relax (x ) = -\frac {5 x^{2}}{12}-\frac {x \left (-x -\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4} \\ y \relax (x ) = -\frac {5 x^{2}}{12}-\frac {x \left (-x +\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4} \\ y \relax (x ) = -\frac {5 x^{2}}{12}+\frac {x \left (x -\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4} \\ y \relax (x ) = -\frac {5 x^{2}}{12}+\frac {x \left (x +\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.719 (sec). Leaf size: 149
DSolve[x^2 - y[x] + 4*x*y'[x] + 3*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{36} \left (-9 x^2+6 x-2 \sqrt {3} \sqrt {e^{2 c_1} (x+1)^2}+3+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{36} \left (-9 x^2+6 x+2 \sqrt {3} \sqrt {e^{2 c_1} (x+1)^2}+3+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{12} \left (-x+1+e^{c_1}\right ) \left (3 x+1+e^{c_1}\right ) \\ y(x)\to -\frac {x^2}{3} \\ y(x)\to \frac {1}{12} ((2-3 x) x+1) \\ \end{align*}