Internal problem ID [9145]
Internal file name [OUTPUT/8080_Monday_June_06_2022_01_41_18_AM_64841268/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 811.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x^{3} {\mathrm e}^{y}+x^{4}+{\mathrm e}^{y} y-{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )+x y-\ln \left ({\mathrm e}^{y}+x \right ) x +x}{x^{2}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{3} {\mathrm e}^{y}+x^{4}-y^{\prime } x^{2}+{\mathrm e}^{y} y-{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )+x y-\ln \left ({\mathrm e}^{y}+x \right ) x +x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-x^{3} {\mathrm e}^{y}-x^{4}+{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )-{\mathrm e}^{y} y+\ln \left ({\mathrm e}^{y}+x \right ) x -x y-x}{x^{2}} \end {array} \]
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: 0.079 (sec). Leaf size: 32
dsolve(diff(y(x),x) = (x^3*exp(y(x))+x^4+exp(y(x))*y(x)-exp(y(x))*ln(exp(y(x))+x)+x*y(x)-ln(exp(y(x))+x)*x+x)/x^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{3}}{2}+c_{1} x +\ln \left (-\frac {x}{-1+{\mathrm e}^{\frac {x \left (x^{2}+2 c_{1} \right )}{2}}}\right ) \]
✓ Solution by Mathematica
Time used: 4.135 (sec). Leaf size: 29
DSolve[y'[x] == (x + E^y[x]*x^3 + x^4 - E^y[x]*Log[E^y[x] + x] - x*Log[E^y[x] + x] + E^y[x]*y[x] + x*y[x])/x^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\log \left (\frac {-1+e^{-\frac {1}{2} x \left (x^2+2 c_1\right )}}{x}\right ) \]