29.28.22 problem 820

Internal problem ID [5403]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 28
Problem number : 820
Date solved : Monday, January 27, 2025 at 11:20:54 AM
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: 0.179 (sec). Leaf size: 43

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

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