Internal problem ID [8144]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 564.
ODE order: 1.
ODE degree: -1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
Solve \begin {gather*} \boxed {\ln \left (y^{\prime }\right )+a \left (x y^{\prime }-y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 36
dsolve(ln(diff(y(x),x))+a*(x*diff(y(x),x)-y(x))=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\ln \left (-\frac {1}{x a}\right )}{a}-\frac {1}{a} \\ y \relax (x ) = x c_{1}+\frac {\ln \left (c_{1}\right )}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 36
DSolve[Log[y'[x]] + a*(-y[x] + x*y'[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\log (c_1)}{a}+c_1 x \\ y(x)\to \frac {\log \left (-\frac {1}{a x}\right )-1}{a} \\ \end{align*}