Internal problem ID [10192]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
25. Equations solvable for \(y\). Page 52
Problem number: Ex 1.
ODE order: 1.
ODE degree: -1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {2 y^{\prime } x -y+\ln \left (y^{\prime }\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve(2*diff(y(x),x)*x-y(x)+ln(diff(y(x),x))=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -1+\sqrt {4 x c_{1}+1}+\ln \left (\frac {-1+\sqrt {4 x c_{1}+1}}{2 x}\right ) \\ y \relax (x ) = -1-\sqrt {4 x c_{1}+1}+\ln \left (-\frac {1+\sqrt {4 x c_{1}+1}}{2 x}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.111 (sec). Leaf size: 32
DSolve[2*y'[x]*x-y[x]+Log[y'[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\text {ProductLog}\left (2 x e^{y(x)}\right )-\log \left (\text {ProductLog}\left (2 x e^{y(x)}\right )+2\right )-y(x)=c_1,y(x)\right ] \]