Internal problem ID [8947]
Internal file name [OUTPUT/7882_Monday_June_06_2022_12_50_11_AM_74244311/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 613.
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.
\[ \boxed {y^{\prime }-\frac {x +y+F \left (-\frac {-y+x \ln \left (x \right )}{x}\right ) x^{2}}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -F \left (-\frac {-y+x \ln \left (x \right )}{x}\right ) x^{2}+y^{\prime } x -y-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x +y+F \left (-\frac {-y+x \ln \left (x \right )}{x}\right ) x^{2}}{x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[1, (x+y)/x]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 34
dsolve(diff(y(x),x) = (x+y(x)+F(-(-y(x)+x*ln(x))/x)*x^2)/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= x \left (\ln \left (x \right )+\operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right )\right ) \\ y \left (x \right ) &= \left (\ln \left (x \right )+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right )\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.233 (sec). Leaf size: 226
DSolve[y'[x] == (x + x^2*F[(-(x*Log[x]) + y[x])/x] + y[x])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x F\left (\frac {K[2]-x \log (x)}{x}\right ) \int _1^x\left (-\frac {K[2] F'\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )^2 K[1]^3}-\frac {F'\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )^2 K[1]^2}+\frac {1}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right ) K[1]^2}\right )dK[1]+1}{x F\left (\frac {K[2]-x \log (x)}{x}\right )}dK[2]+\int _1^x\left (\frac {y(x)}{F\left (\frac {y(x)-K[1] \log (K[1])}{K[1]}\right ) K[1]^2}+\frac {1}{F\left (\frac {y(x)-K[1] \log (K[1])}{K[1]}\right ) K[1]}+1\right )dK[1]=c_1,y(x)\right ] \]