24.71 problem 71

Internal problem ID [10807]
Internal file name [OUTPUT/9755_Wednesday_June_08_2022_05_54_27_PM_62639740/index.tex]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 71.
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.

\[ \boxed {y y^{\prime }-{\mathrm e}^{a x} \left (2 a \,x^{2}+b +2 x \right ) y={\mathrm e}^{2 a x} \left (-x^{4} a -b \,x^{2}+c \right )} \] Unable to determine ODE type.

24.71.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-{\mathrm e}^{a x} \left (2 a \,x^{2}+b +2 x \right ) y={\mathrm e}^{2 a x} \left (-x^{4} a -b \,x^{2}+c \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{a x} \left (2 a \,x^{2}+b +2 x \right ) y+{\mathrm e}^{2 a x} \left (-x^{4} a -b \,x^{2}+c \right )}{y} \end {array} \]

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)*(2*a^2*x^2+a*b+6*a*x+2)/(2*a*x^2+b+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)+2*y(x)*(a^2*x^4+a*b*x^2+2*a*x^3-a*c+b*x)/(a*x^4+b*x^2-c), y(x)`      *** Sublevel 
      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)*(2*a^2*x^2+a*b+6*a*x+2)/(2*a*x^2+b+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)+(1/4)*K[1]/x, y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE`, diff(y(x), x)-c*K[1]*(a*b-2)/(x*b^3), y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE`, diff(y(x), x)+(3/4)*a*b*K[1]/(x*(3*a*b+1)), y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE`, diff(y(x), x)+(1/3)*K[1]*(a*c+b^2)/(x*b^2), y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE`, diff(y(x), x)-(1/6)*K[1]*(2*a^2*c-a*b^2-2*b)/(x*b*(a*b+2)), y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature 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)*diff(y(x),x)=exp(a*x)*(2*a*x^2+2*x+b)*y(x)+exp(2*a*x)*(-a*x^4-b*x^2+c),y(x), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

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

DSolve[y[x]*y'[x]==Exp[a*x]*(2*a*x^2+2*x+b)*y[x]+Exp[2*a*x]*(-a*x^4-b*x^2+c),y[x],x,IncludeSingularSolutions -> True]
 

Not solved