Internal problem ID [8456]
Internal file name [OUTPUT/7389_Sunday_June_05_2022_10_54_15_PM_94109174/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 120.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {x y^{\prime }-y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right )}{x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[1, 2*y/x]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 16
dsolve(x*diff(y(x),x) - y(x)*(x*ln(x^2/y(x))+2)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{2} {\mathrm e}^{-{\mathrm e}^{-x} c_{1}} \]
✓ Solution by Mathematica
Time used: 0.25 (sec). Leaf size: 20
DSolve[x*y'[x] - y[x]*(x*Log[x^2/y[x]]+2)==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^2 e^{-2 c_1 e^{-x}} \]