Internal problem ID [8738]
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`]]
\[ \boxed {3 {y^{\prime }}^{2}+4 y^{\prime } x -y=-x^{2}} \]
✓ Solution by Maple
Time used: 0.094 (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 \left (x \right ) = -\frac {x^{2}}{3} y \left (x \right ) = -\frac {5 x^{2}}{12}-\frac {\left (-x -\sqrt {3}\, c_{1} \right ) x}{6}+\frac {c_{1}^{2}}{4} y \left (x \right ) = -\frac {5 x^{2}}{12}-\frac {\left (-x +\sqrt {3}\, c_{1} \right ) x}{6}+\frac {c_{1}^{2}}{4} y \left (x \right ) = -\frac {5 x^{2}}{12}+\frac {\left (x -\sqrt {3}\, c_{1} \right ) x}{6}+\frac {c_{1}^{2}}{4} y \left (x \right ) = -\frac {5 x^{2}}{12}+\frac {\left (x +\sqrt {3}\, c_{1} \right ) x}{6}+\frac {c_{1}^{2}}{4} \end{align*}
✓ Solution by Mathematica
Time used: 3.78 (sec). Leaf size: 121
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}{12} \left (-3 x^2+2 x-2 e^{c_1} (x+1)+1+e^{2 c_1}\right ) y(x)\to \frac {-3 x^2-3 x^2 \tanh ^2\left (\frac {c_1}{2}\right )+4 x+2 (3 x-2) x \tanh \left (\frac {c_1}{2}\right )+4}{12 \left (-1+\tanh \left (\frac {c_1}{2}\right )\right ){}^2} y(x)\to -\frac {x^2}{3} y(x)\to \frac {1}{12} \left (-3 x^2+2 x+1\right ) \end{align*}