Internal problem ID [10754]
Internal file name [OUTPUT/9702_Monday_June_06_2022_03_29_21_PM_54352336/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: 18.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {3 y y^{\prime }-\frac {\left (-7 \lambda s \left (3 s +4 \lambda \right ) x +6 s -2 \lambda \right ) y}{x^{\frac {1}{3}}}=\frac {6 \lambda s x -6}{x^{\frac {2}{3}}}+2 \left (\lambda s \left (3 s +4 \lambda \right ) x +5 \lambda \right ) \left (-\lambda s \left (3 s +4 \lambda \right ) x +3 s +4 \lambda \right ) x^{\frac {1}{3}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 32 x^{3} \lambda ^{4} s^{2}+48 x^{3} \lambda ^{3} s^{3}+18 x^{3} \lambda ^{2} s^{4}+8 x^{2} \lambda ^{3} s -18 x^{2} \lambda ^{2} s^{2}-18 x^{2} \lambda \,s^{3}+28 y x^{\frac {4}{3}} \lambda ^{2} s +21 y x^{\frac {4}{3}} \lambda \,s^{2}-40 x \,\lambda ^{2}-36 \lambda s x +3 y y^{\prime } x^{\frac {2}{3}}+2 y x^{\frac {1}{3}} \lambda -6 y x^{\frac {1}{3}} s +6=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-32 x^{3} \lambda ^{4} s^{2}-48 x^{3} \lambda ^{3} s^{3}-18 x^{3} \lambda ^{2} s^{4}-28 y x^{\frac {4}{3}} \lambda ^{2} s -21 y x^{\frac {4}{3}} \lambda \,s^{2}-8 x^{2} \lambda ^{3} s +18 x^{2} \lambda ^{2} s^{2}+18 x^{2} \lambda \,s^{3}-2 y x^{\frac {1}{3}} \lambda +6 y x^{\frac {1}{3}} s +40 x \,\lambda ^{2}+36 \lambda s x -6}{3 y x^{\frac {2}{3}}} \end {array} \]
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)+(2/3)*y(x)*(28*lambda^2*s*x+21*lambda*s^2*x-lambda+3*s)/(x*(28*lambda^2*s*x+21*la 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)+(1/3)*y(x)*(112*lambda^4*s^2*x^3+168*lambda^3*s^3*x^3+63*lambda^2*s^4*x^3+16*lamb 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)+(1/9)*(-x^2*lambda+3*s*x^2+9*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)+(1/3)*(28*lambda*s*x^2+21*s^2*x^2+30*lambda*y(x)+27*y(x)*s)/((10*lambda+9*s)*x), 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(3*y(x)*diff(y(x),x)=(-7*lambda*s*(3*s+4*lambda)*x+6*s-2*lambda)*x^(-1/3)*y(x)+6*(lambda*s*x-1)*x^(-2/3)+2*(lambda*s*(3*s+4*lambda)*x+5*lambda)*(-lambda*s*(3*s+4*lambda)*x+3*s+4*lambda)*x^(1/3),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[3*y[x]*y'[x]==(-7*\[Lambda]*s*(3*s+4*\[Lambda])*x+6*s-2*\[Lambda])*x^(-1/3)*y[x]+6*(\[Lambda]*s*x-1)*x^(-2/3)+2*(\[Lambda]*s*(3*s+4*\[Lambda])*x+5*\[Lambda])*(-\[Lambda]*s*(3*s+4*\[Lambda])*x+3*s+4*\[Lambda])*x^(1/3),y[x],x,IncludeSingularSolutions -> True]
Timed out