Internal problem ID [8916]
Internal file name [OUTPUT/7851_Monday_June_06_2022_12_47_04_AM_16330371/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 582.
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.
\[ \boxed {y^{\prime }-\frac {1+F \left (\frac {a x y+1}{a x}\right ) a \,x^{2}}{a \,x^{2}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & F \left (\frac {a x y+1}{a x}\right ) a \,x^{2}-y^{\prime } a \,x^{2}+1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-1-F \left (\frac {a x y+1}{a x}\right ) a \,x^{2}}{a \,x^{2}} \end {array} \]
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`[1, 1/a/x^2]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 48
dsolve(diff(y(x),x) = (1+F((y(x)*a*x+1)/a/x)*a*x^2)/a/x^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right ) a x -1}{a x} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) a x -1}{a x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.292 (sec). Leaf size: 142
DSolve[y'[x] == (1 + a*x^2*F[(1 + a*x*y[x])/(a*x)])/(a*x^2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {a x K[2]+1}{a x}\right ) \int _1^x\frac {F'\left (\frac {a K[1] K[2]+1}{a K[1]}\right )}{a F\left (\frac {a K[1] K[2]+1}{a K[1]}\right )^2 K[1]^2}dK[1]-1}{F\left (\frac {a x K[2]+1}{a x}\right )}dK[2]+\int _1^x\left (-1-\frac {1}{a K[1]^2 F\left (\frac {a K[1] y(x)+1}{a K[1]}\right )}\right )dK[1]=c_1,y(x)\right ] \]