60.1.391 problem 392

Internal problem ID [10405]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 392
Date solved : Monday, January 27, 2025 at 07:40:03 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} {y^{\prime }}^{2}-x y y^{\prime }+y^{2} \ln \left (a y\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.163 (sec). Leaf size: 43

dsolve(diff(y(x),x)^2-x*y(x)*diff(y(x),x)+y(x)^2*ln(a*y(x)) = 0,y(x), singsol=all)
 

\begin{align*} y &= \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{a} \\ y &= \frac {{\mathrm e}^{c_{1} \left (x -c_{1} \right )}}{a} \\ y &= \frac {{\mathrm e}^{-c_{1} \left (x +c_{1} \right )}}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.303 (sec). Leaf size: 30

DSolve[Log[a*y[x]]*y[x]^2 - x*y[x]*D[y[x],x] + D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{\frac {1}{4} c_1 (2 x-c_1)}}{a} \\ y(x)\to 0 \\ \end{align*}