Internal problem ID [9203]
Internal file name [OUTPUT/8139_Monday_June_06_2022_01_53_36_AM_83190910/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 870.
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(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x +x^{3}+x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -{\mathrm e}^{\frac {y}{x}} x^{4}-{\mathrm e}^{\frac {y}{x}} x^{3}-{\mathrm e}^{-\frac {y}{x}} y \,{\mathrm e}^{\frac {y}{x}}-{\mathrm e}^{-\frac {y}{x}} {\mathrm e}^{\frac {y}{x}} x +y^{\prime } x -{\mathrm e}^{\frac {y}{x}} x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{\frac {y}{x}} x^{4}+{\mathrm e}^{\frac {y}{x}} x^{3}+{\mathrm e}^{-\frac {y}{x}} y \,{\mathrm e}^{\frac {y}{x}}+{\mathrm e}^{-\frac {y}{x}} {\mathrm e}^{\frac {y}{x}} x +{\mathrm e}^{\frac {y}{x}} x}{x} \end {array} \]
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 `, `-> Computing symmetries using: way = 5`[0, exp(y/x)]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
dsolve(diff(y(x),x) = (exp(-y(x)/x)*y(x)+exp(-y(x)/x)*x+x+x^3+x^4)*exp(y(x)/x)/x,y(x), singsol=all)
\[ y \left (x \right ) = \left (2 \ln \left (2\right )+\ln \left (3\right )-\ln \left (\frac {-3 x^{4}-4 x^{3}-12 c_{1} -12 x}{x}\right )\right ) x \]
✓ Solution by Mathematica
Time used: 4.311 (sec). Leaf size: 32
DSolve[y'[x] == (E^(y[x]/x)*(x + x/E^(y[x]/x) + x^3 + x^4 + y[x]/E^(y[x]/x)))/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -x \log \left (-\frac {x^3}{4}-\frac {x^2}{3}-\frac {c_1}{x}-1\right ) \]