Internal problem ID [9136]
Internal file name [OUTPUT/8071_Monday_June_06_2022_01_39_30_AM_94180418/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 802.
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)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -x^{2} y^{\prime }+\textit {\_F1} \left (\frac {y x +1}{x}\right ) x +1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-\textit {\_F1} \left (\frac {y x +1}{x}\right ) x -1}{x^{2}} \end {array} \]
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`[x, 1/x]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 41
dsolve(diff(y(x),x) = -(-1/x-_F1(y(x)+1/x))/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\operatorname {RootOf}\left (f_{1} \left (\textit {\_Z} \right )\right ) x -1}{x} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x -1}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.194 (sec). Leaf size: 96
DSolve[y'[x] == (x^(-1) + F1[x^(-1) + y[x]])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}\left (K[2]+\frac {1}{x}\right ) \int _1^x-\frac {\text {F1}'\left (K[2]+\frac {1}{K[1]}\right )}{\text {F1}\left (K[2]+\frac {1}{K[1]}\right )^2 K[1]^2}dK[1]+1}{\text {F1}\left (K[2]+\frac {1}{x}\right )}dK[2]+\int _1^x\left (\frac {1}{K[1]}+\frac {1}{K[1]^2 \text {F1}\left (y(x)+\frac {1}{K[1]}\right )}\right )dK[1]=c_1,y(x)\right ] \]