Internal problem ID [8555]
Internal file name [OUTPUT/7488_Sunday_June_05_2022_11_01_21_PM_40300235/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 219.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {\left (y+g \left (x \right )\right ) y^{\prime }-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y=f_{0} \left (x \right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y+g \left (x \right )\right ) y^{\prime }-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y=f_{0} \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {f_{2} \left (x \right ) y^{2}+f_{1} \left (x \right ) y+f_{0} \left (x \right )}{y+g \left (x \right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel Looking for potential symmetries Looking for potential symmetries Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x) = -y(x)*(diff(f__2(x), x))/f__2(x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+y(x)*(diff(f__2(x), x))/f__2(x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-(y(x)*f__0(x)*(diff(g(x), x))-K[1]*f__1(x)*g(x)-y(x)*(diff(f__0(x), x))*g(x)+K[1] Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(y(x)*(diff(f__2(x), x))*g(x)-y(x)*f__2(x)*(diff(g(x), x))+f__2(x)*K[1]+y(x)*(dif Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(2*f__2(x)*g(x)*K[1]+y(x)*(diff(f__1(x), x))*g(x)-y(x)*f__1(x)*(diff(g(x), x))+y( Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-(K[1]*f__0(x)*g(x)+y(x)*f__0(x)*(diff(g(x), x))-y(x)*(diff(f__0(x), x))*g(x))/(f_ Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-(-y(x)*(diff(f__1(x), x))*g(x)+y(x)*f__1(x)*(diff(g(x), x))+2*K[1]*f__0(x)-y(x)*( Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(y(x)*(diff(f__2(x), x))*g(x)+f__2(x)*g(x)*K[1]-y(x)*f__2(x)*(diff(g(x), x))+y(x) Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)-(y(x)*f__0(x)*(diff(g(x), x))-y(x)*(diff(f__0(x), x))*g(x)-K[1]*f__1(x)*g(x)+K[1] Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(y(x)*(diff(f__2(x), x))*g(x)-y(x)*f__2(x)*(diff(g(x), x))+y(x)*(diff(f__1(x), x) Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(y(x)*(diff(f__1(x), x))*g(x)-y(x)*f__1(x)*(diff(g(x), x))+2*f__2(x)*g(x)*K[1]+y( Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+y(x)*(g(x)*(diff(f__2(x), x))-(diff(g(x), x))*f__2(x)+diff(f__1(x), x))/(f__2(x)* Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-(y(x)*f__0(x)*(diff(g(x), x))-y(x)*(diff(f__0(x), x))*g(x)-3*f__2(x)*g(x)*K[1]-f_ Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(y(x)*(diff(f__1(x), x))*g(x)-y(x)*f__1(x)*(diff(g(x), x))+y(x)*(diff(f__0(x), x) Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> trying a symmetry pattern of the form [F(x)+G(y), 0] -> trying a symmetry pattern of the form [0, F(x)+G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying a symmetry pattern of conformal type`
✗ Solution by Maple
dsolve((y(x)+g(x))*diff(y(x),x)-f__2(x)*y(x)^2-f__1(x)*y(x)-f__0(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(y[x]+g[x])*y'[x]-f2[x]*y[x]^2-f1[x]*y[x]-f0[x]==0,y[x],x,IncludeSingularSolutions -> True]
Timed out