Internal problem ID [3843]
Internal file name [OUTPUT/3336_Sunday_June_05_2022_09_09_38_AM_28303486/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 21
Problem number: 593.
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.
\[ \boxed {\left (\operatorname {g0} \left (x \right )+y \operatorname {g1} \left (x \right )\right ) y^{\prime }-\operatorname {f1} \left (x \right ) y-\operatorname {f2} \left (x \right ) y^{2}-\operatorname {f3} \left (x \right ) y^{3}=\operatorname {f0} \left (x \right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\mathit {g0} \left (x \right )+y \mathit {g1} \left (x \right )\right ) y^{\prime }-\mathit {f1} \left (x \right ) y-\mathit {f2} \left (x \right ) y^{2}-\mathit {f3} \left (x \right ) y^{3}=\mathit {f0} \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\mathit {f0} \left (x \right )+\mathit {f1} \left (x \right ) y+\mathit {f2} \left (x \right ) y^{2}+\mathit {f3} \left (x \right ) y^{3}}{\mathit {g0} \left (x \right )+y \mathit {g1} \left (x \right )} \end {array} \]
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)*(g1(x)*(diff(f3(x), x))-f3(x)*(diff(g1(x), x)))/(g1(x)*f3(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(f0(x), x))*g0(x)+K[1]*g0(x)*f1(x)-K[1]*g1(x)*f0(x)-y(x)*f0(x)*(diff(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)+(y(x)*g0(x)*(diff(f3(x), x))+2*g1(x)*f3(x)*K[1]+y(x)*g1(x)*(diff(f2(x), 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(f0(x), x))*g1(x)+y(x)*(diff(f1(x), x))*g0(x)+2*g0(x)*f2(x)*K[1]-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(f1(x), x))*g1(x)+3*g0(x)*f3(x)*K[1]+y(x)*g0(x)*(diff(f2(x), x))+g1(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(f0(x), x))*g0(x)-K[1]*g0(x)*f0(x)-y(x)*f0(x)*(diff(g0(x), x)))/(g0(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)+(g1(x)*f3(x)*K[1]+y(x)*g1(x)*(diff(f3(x), x))-y(x)*f3(x)*(diff(g1(x), x)))/(g1(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*g0(x)*f3(x)*K[1]+y(x)*g0(x)*(diff(f3(x), x))+y(x)*g1(x)*(diff(f2(x), 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(f0(x), x))*g1(x)+y(x)*(diff(f1(x), x))*g0(x)-2*K[1]*g1(x)*f0(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(f1(x), x))*g1(x)+g0(x)*f2(x)*K[1]+y(x)*g0(x)*(diff(f2(x), x))-K[1]*f1 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)*(diff(f0(x), x))*g0(x)+K[1]*g0(x)*f1(x)-K[1]*g1(x)*f0(x)-y(x)*f0(x)*(diff(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)+(y(x)*g0(x)*(diff(f3(x), x))+2*g1(x)*f3(x)*K[1]+y(x)*g1(x)*(diff(f2(x), 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(f0(x), x))*g1(x)+y(x)*(diff(f1(x), x))*g0(x)+2*g0(x)*f2(x)*K[1]-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(f1(x), x))*g1(x)+3*g0(x)*f3(x)*K[1]+y(x)*g0(x)*(diff(f2(x), x))+g1(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(f0(x), x))*g0(x)+3*g0(x)*f2(x)*K[1]+K[1]*f1(x)*g1(x)-y(x)*f0(x)*(diff 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)*(g0(x)*(diff(f3(x), x))-(diff(g0(x), x))*f3(x)+g1(x)*(diff(f2(x), x))-(diff( 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(f1(x), x))*g1(x)+y(x)*g0(x)*(diff(f2(x), x))+3*g1(x)*f3(x)*K[1]-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(f0(x), x))*g1(x)+y(x)*(diff(f1(x), x))*g0(x)+4*g0(x)*f3(x)*K[1]+2*g1( 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((g0(x)+y(x)*g1(x))*diff(y(x),x) = f0(x)+f1(x)*y(x)+f2(x)*y(x)^2+f3(x)*y(x)^3,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(g0[x]+y[x] g1[x])y'[x]==f0[x]+f1[x] y[x]+f2[x] y[x]^2+f3[x] y[x]^3,y[x],x,IncludeSingularSolutions -> True]
Timed out