24.78 problem 78

Internal problem ID [10814]
Internal file name [OUTPUT/9762_Thursday_June_09_2022_12_57_51_AM_85664660/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: 78.
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.

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

24.78.1 Maple step by step solution

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

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)+2*y(x)*(2*ln(x)+1)/(x*(2*ln(x)^2+2*ln(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)*(ln(x)^4+4*ln(x)^3+a*ln(x)^2+2*ln(x)*a-b)/(x*(ln(x)^4+a*ln(x)^2-b)), 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) = -2*y(x)*(2*ln(x)+1)/(x*(2*ln(x)^2+2*ln(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)+(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)-K[1]*b*(a-2)/(x*a^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*K[1]/(x*(3*a+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^2+b)/(x*a^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]*(a^2+2*a-2*b)/(x*a*(a+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)=(2*(ln(x))^2+2*ln(x)+a)*y(x)+x*(- (ln(x))^4-a*(ln(x))^2+b),y(x), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

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

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

Not solved