Internal problem ID [9195]
Internal file name [OUTPUT/8130_Monday_June_06_2022_01_51_41_AM_37740741/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 861.
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)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-f_{1} \left ({\mathrm e}^{\frac {1}{x}} y\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }-f_{1} \left (\frac {y}{{\mathrm e}^{-\frac {1}{x}}}\right ) x \,{\mathrm e}^{-\frac {1}{x}}-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {f_{1} \left (\frac {y}{{\mathrm e}^{-\frac {1}{x}}}\right ) x \,{\mathrm e}^{-\frac {1}{x}}+y}{x^{2}} \end {array} \]
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, y/x]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 39
dsolve(diff(y(x),x) = -(-1/x*y(x)/exp(-1/x)-_F1(y(x)/exp(-1/x)))*exp(-1/x)/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (f_{1} \left (\textit {\_Z} \right )\right ) {\mathrm e}^{-\frac {1}{x}} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) {\mathrm e}^{-\frac {1}{x}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.915 (sec). Leaf size: 152
DSolve[y'[x] == (F1[E^x^(-1)*y[x]] + (E^x^(-1)*y[x])/x)/(E^x^(-1)*x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}\left (e^{\frac {1}{x}} K[2]\right ) \int _1^x\left (\frac {e^{\frac {1}{K[1]}}}{\text {F1}\left (e^{\frac {1}{K[1]}} K[2]\right ) K[1]^2}-\frac {e^{\frac {2}{K[1]}} K[2] \text {F1}'\left (e^{\frac {1}{K[1]}} K[2]\right )}{\text {F1}\left (e^{\frac {1}{K[1]}} K[2]\right )^2 K[1]^2}\right )dK[1]+e^{\frac {1}{x}}}{\text {F1}\left (e^{\frac {1}{x}} K[2]\right )}dK[2]+\int _1^x\left (\frac {e^{\frac {1}{K[1]}} y(x)}{\text {F1}\left (e^{\frac {1}{K[1]}} y(x)\right ) K[1]^2}+\frac {1}{K[1]}\right )dK[1]=c_1,y(x)\right ] \]