Internal problem ID [7608]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 27.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }+a y \left (y-x \right )-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.101 (sec). Leaf size: 71
dsolve(diff(y(x),x) + a*y(x)*(y(x)-x) - 1=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\sqrt {\pi }\, \sqrt {2}\, \erf \left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) a x +2 a^{\frac {3}{2}} c_{1} x +2 \sqrt {a}\, {\mathrm e}^{-\frac {a \,x^{2}}{2}}}{\sqrt {\pi }\, \erf \left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) \sqrt {2}\, a +2 a^{\frac {3}{2}} c_{1}} \]
✓ Solution by Mathematica
Time used: 3.779 (sec). Leaf size: 59
DSolve[y'[x] + a*y[x]*(y[x]-x) - 1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x+\frac {2 c_1 e^{-\frac {a x^2}{2}}}{\sqrt {a} \left (2 \sqrt {a}+\sqrt {2 \pi } c_1 \operatorname {Erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )\right )} \\ \end{align*}