Internal problem ID [4059]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 28
Problem number: 820.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\[ \boxed {{y^{\prime }}^{2}-x y y^{\prime }+y^{2} \ln \left (y a \right )=0} \]
✓ Solution by Maple
Time used: 0.203 (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 (c_{1} +x \right )}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.324 (sec). Leaf size: 30
DSolve[(y'[x])^2-x y'[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*}