Internal problem ID [8945]
Internal file name [OUTPUT/7880_Monday_June_06_2022_12_50_00_AM_44781940/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 611.
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.
\[ \boxed {y^{\prime }-\frac {-2 x -y+F \left (\left (y+x \right ) x \right )}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x +y-F \left (\left (y+x \right ) x \right )+2 x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 x -y+F \left (\left (y+x \right ) x \right )}{x} \end {array} \]
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, -(y+2*x)/x]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 44
dsolve(diff(y(x),x) = (-2*x-y(x)+F((x+y(x))*x))/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {-x^{2}+\operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right )}{x} \\ y \left (x \right ) &= \frac {-x^{2}+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right )}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.296 (sec). Leaf size: 191
DSolve[y'[x] == (-2*x + F[x*(x + y[x])] - y[x])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x+F(x (x+K[2])) \int _1^x\left (\frac {2 F'(K[1] (K[1]+K[2])) K[1]^2}{F(K[1] (K[1]+K[2]))^2}+\frac {(K[2]-F(K[1] (K[1]+K[2]))) F'(K[1] (K[1]+K[2])) K[1]}{F(K[1] (K[1]+K[2]))^2}-\frac {1-K[1] F'(K[1] (K[1]+K[2]))}{F(K[1] (K[1]+K[2]))}\right )dK[1]}{F(x (x+K[2]))}dK[2]+\int _1^x\left (-\frac {2 K[1]}{F(K[1] (K[1]+y(x)))}-\frac {y(x)-F(K[1] (K[1]+y(x)))}{F(K[1] (K[1]+y(x)))}\right )dK[1]=c_1,y(x)\right ] \]