Internal problem ID [8057]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 477.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _rational, _dAlembert]
Solve \begin {gather*} \boxed {a y \left (y^{\prime }\right )^{2}+\left (2 x -b \right ) y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.656 (sec). Leaf size: 933
dsolve(a*y(x)*diff(y(x),x)^2+(2*x-b)*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {-2 x +b}{2 \sqrt {-a}} \\ y \relax (x ) = \frac {-2 x +b}{2 \sqrt {-a}} \\ y \relax (x ) = 0 \\ \int _{\textit {\_b}}^{x}\frac {-4 \textit {\_a} +2 b +2 \sqrt {4 a y \relax (x )^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}{4 a y \relax (x )^{2}+\sqrt {4 a y \relax (x )^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -2 \sqrt {4 a y \relax (x )^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} +b^{2}-4 b \textit {\_a} +4 \textit {\_a}^{2}}d \textit {\_a} +\int _{}^{y \relax (x )}\left (-\frac {4 a \textit {\_f}}{4 a \,\textit {\_f}^{2}+\sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}\, b -2 \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}\, x +b^{2}-4 b x +4 x^{2}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {8 a \textit {\_f}}{\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \left (4 a \,\textit {\_f}^{2}+\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} +b^{2}-4 b \textit {\_a} +4 \textit {\_a}^{2}\right )}-\frac {2 \left (-2 \textit {\_a} +b +\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\right ) \left (8 a \textit {\_f} +\frac {4 b a \textit {\_f}}{\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}-\frac {8 \textit {\_a} a \textit {\_f}}{\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}\right )}{\left (4 a \,\textit {\_f}^{2}+\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} +b^{2}-4 b \textit {\_a} +4 \textit {\_a}^{2}\right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \\ \int _{\textit {\_b}}^{x}\frac {4 \textit {\_a} -2 b +2 \sqrt {4 a y \relax (x )^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}{-4 a y \relax (x )^{2}+\sqrt {4 a y \relax (x )^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -2 \sqrt {4 a y \relax (x )^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} -b^{2}+4 b \textit {\_a} -4 \textit {\_a}^{2}}d \textit {\_a} +\int _{}^{y \relax (x )}\left (\frac {4 a \textit {\_f}}{-4 a \,\textit {\_f}^{2}+\sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}\, b -2 \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}\, x -b^{2}+4 b x -4 x^{2}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {8 a \textit {\_f}}{\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \left (-4 a \,\textit {\_f}^{2}+\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} -b^{2}+4 b \textit {\_a} -4 \textit {\_a}^{2}\right )}-\frac {2 \left (2 \textit {\_a} -b +\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\right ) \left (-8 a \textit {\_f} +\frac {4 b a \textit {\_f}}{\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}-\frac {8 \textit {\_a} a \textit {\_f}}{\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}}\right )}{\left (-4 a \,\textit {\_f}^{2}+\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, b -2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 b \textit {\_a} +b^{2}}\, \textit {\_a} -b^{2}+4 b \textit {\_a} -4 \textit {\_a}^{2}\right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.839 (sec). Leaf size: 187
DSolve[-y[x] + (-b + 2*x)*y'[x] + a*y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\ y(x)\to \sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\ y(x)\to -\frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\ y(x)\to \frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\ y(x)\to 0 \\ y(x)\to -\frac {i (b-2 x)}{2 \sqrt {a}} \\ y(x)\to \frac {i (b-2 x)}{2 \sqrt {a}} \\ \end{align*}