Internal problem ID [15108]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number: 220.
ODE order: 1.
ODE degree: 0.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {y-2 y^{\prime } x -\ln \left (y^{\prime }\right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 69
dsolve(y(x)=2*x*diff(y(x),x)+ln(diff(y(x),x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -1+\sqrt {4 c_{1} x +1}-\ln \left (2\right )+\ln \left (\frac {-1+\sqrt {4 c_{1} x +1}}{x}\right ) \\ y \left (x \right ) &= -1-\sqrt {4 c_{1} x +1}-\ln \left (2\right )+\ln \left (\frac {-1-\sqrt {4 c_{1} x +1}}{x}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.096 (sec). Leaf size: 32
DSolve[y[x]==2*x*y'[x]+Log[y'[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [W\left (2 x e^{y(x)}\right )-\log \left (W\left (2 x e^{y(x)}\right )+2\right )-y(x)=c_1,y(x)\right ] \]