1.249 problem 250

Internal problem ID [8586]
Internal file name [OUTPUT/7519_Sunday_June_05_2022_11_02_49_PM_80314908/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 250.
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 B`]]

Unable to solve or complete the solution.

\[ \boxed {\left (B x y+A \,x^{2}+x a +b y+c \right ) y^{\prime }+A x y+\beta y=B g \left (x \right )^{2}-x \alpha -\gamma } \] Unable to determine ODE type.

1.249.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (B x y+A \,x^{2}+x a +b y+c \right ) y^{\prime }+A x y+\beta y=B g \left (x \right )^{2}-x \alpha -\gamma \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {B g \left (x \right )^{2}-A x y-x \alpha -\beta y-\gamma }{B x y+A \,x^{2}+x a +b y+c} \end {array} \]

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) = 0, 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) = -B*y(x)/(B*x+a), 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)*(A*b-B*beta)/((B*x+b)*(A*x+beta)), 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)*(2*B^2*g(x)*x*(diff(g(x), x))+A^2*x^2-B^2*g(x)^2+2*B*g(x)*b*(diff(g(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*A*B*y(x)*g(x)*(diff(g(x), x))*x^2+A^2*K[1]*x^3+2*A*B*y(x)*g(x)^2*x+B^2*K[1]*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)*(A*b-B*beta)/((B*x+b)*(A*x+beta)), y(x)`      *** Sublevel 2 *** 
      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((B*x*y(x)+A*x^2+a*x+b*y(x)+c)*diff(y(x),x)-B*g(x)^2+A*x*y(x)+alpha*x+beta*y(x)+gamma=0,y(x), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[(B*x*y[x]+A*x^2+a*x+b*y[x]+c)*y'[x]-B*g[x]^2+A*x*y[x]+\[Alpha]*x+\[Beta]*y[x]+\[Gamma]==0,y[x],x,IncludeSingularSolutions -> True]
 

Timed out