Internal problem ID [8605]
Internal file name [OUTPUT/7538_Sunday_June_05_2022_11_04_07_PM_8226182/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 269.
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 C`]]
Unable to solve or complete the solution.
\[ \boxed {\left (g_{1} \left (x \right ) y+g_{0} \left (x \right )\right ) y^{\prime }-f_{1} \left (x \right ) y-f_{2} \left (x \right ) y^{2}-f_{3} \left (x \right ) y^{3}=f_{0} \left (x \right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (g_{1} \left (x \right ) y+g_{0} \left (x \right )\right ) y^{\prime }-f_{1} \left (x \right ) y-f_{2} \left (x \right ) y^{2}-f_{3} \left (x \right ) y^{3}=f_{0} \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {f_{3} \left (x \right ) y^{3}+f_{2} \left (x \right ) y^{2}+f_{1} \left (x \right ) y+f_{0} \left (x \right )}{g_{1} \left (x \right ) y+g_{0} \left (x \right )} \end {array} \]
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)/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)*(g__1(x)*(diff(f__3(x), x))-f__3(x)*(diff(g__1(x), x)))/(g__1(x)*f__3(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)-(g__1(x)*f__0(x)*K[1]-g__0(x)*f__1(x)*K[1]-g__0(x)*y(x)*(diff(f__0(x), x))+f__0(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)+(g__1(x)*y(x)*(diff(f__0(x), x))+2*g__0(x)*f__2(x)*K[1]+g__0(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)+(2*g__1(x)*f__3(x)*K[1]+g__1(x)*(diff(f__2(x), x))*y(x)+g__0(x)*y(x)*(diff(f__3(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)+(g__1(x)*f__2(x)*K[1]+g__1(x)*(diff(f__1(x), x))*y(x)+3*g__0(x)*f__3(x)*K[1]+g__0 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__0(x)-g__0(x)*y(x)*(diff(f__0(x), x))+f__0(x)*y(x)*(diff(g__0(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)+(g__1(x)*f__3(x)*K[1]+y(x)*g__1(x)*(diff(f__3(x), x))-y(x)*f__3(x)*(diff(g__1(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)-(2*g__1(x)*f__0(x)*K[1]-g__1(x)*y(x)*(diff(f__0(x), x))-g__0(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)+(g__1(x)*(diff(f__2(x), x))*y(x)+2*g__0(x)*f__3(x)*K[1]+g__0(x)*y(x)*(diff(f__3(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)+(g__1(x)*(diff(f__1(x), x))*y(x)-g__1(x)*f__1(x)*K[1]+g__0(x)*f__2(x)*K[1]+g__0(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)+(g__0(x)*y(x)*(diff(f__0(x), x))-f__0(x)*y(x)*(diff(g__0(x), x))-g__1(x)*f__0(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)+(g__1(x)*y(x)*(diff(f__0(x), x))+g__0(x)*(diff(f__1(x), x))*y(x)-f__0(x)*y(x)*(di 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)+(g__1(x)*(diff(f__2(x), x))*y(x)+g__0(x)*y(x)*(diff(f__3(x), x))-f__3(x)*y(x)*(di 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)+(g__1(x)*(diff(f__1(x), x))*y(x)+g__0(x)*(diff(f__2(x), x))*y(x)-f__2(x)*y(x)*(di 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)+(g__0(x)*y(x)*(diff(f__0(x), x))-f__0(x)*y(x)*(diff(g__0(x), x))+g__1(x)*f__1(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)*(g__0(x)*(diff(f__3(x), x))-(diff(g__0(x), x))*f__3(x)+(diff(f__2(x), x))*g_ 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)+(g__1(x)*(diff(f__1(x), x))*y(x)+g__0(x)*(diff(f__2(x), x))*y(x)-f__2(x)*y(x)*(di 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)+(g__1(x)*y(x)*(diff(f__0(x), x))+g__0(x)*(diff(f__1(x), x))*y(x)-f__0(x)*y(x)*(di 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((g__1(x)*y(x)+g__0(x))*diff(y(x),x)-f__1(x)*y(x)-f__2(x)*y(x)^2-f__3(x)*y(x)^3-f__0(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(g1[x]*y[x]+g0[x])*y'[x]-f1[x]*y[x]-f2[x]*y[x]^2-f3[x]*y[x]^3-f0[x]==0,y[x],x,IncludeSingularSolutions -> True]
Timed out