2.250 problem 826

Internal problem ID [9805]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 826
Date solved : Thursday, October 17, 2024 at 10:10:44 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solve

\begin{align*} y^{\prime }&=\frac {\left (3 x y^{2}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )} \end{align*}

Unknown ode type.

2.250.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (3 x y \left (x \right )^{2}+x +3 y \left (x \right )^{2}\right ) y \left (x \right )}{\left (6 y \left (x \right )^{2}+x \right ) x \left (x +1\right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (3 x y \left (x \right )^{2}+x +3 y \left (x \right )^{2}\right ) y \left (x \right )}{\left (6 y \left (x \right )^{2}+x \right ) x \left (x +1\right )} \end {array} \]

2.250.2 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 
Looking for potential symmetries 
trying inverse_Riccati 
trying an equivalence to an Abel ODE 
equivalence obtained to this Abel ODE: diff(y(x),x) = 2/x/(x+1)*y(x)+12*(x-1)/x^2/(x+1)*y(x)^2-36*(x-1)/x^3/(x+1)*y(x)^3 
trying to solve the Abel ODE ... 
<- Abel successful 
equivalence to an Abel ODE successful, Abel ODE has been solved`
 

2.250.3 Maple dsolve solution

Solving time : 0.003 (sec)
Leaf size : 59

dsolve(diff(y(x),x) = 1/(6*y(x)^2+x)*(3*x*y(x)^2+x+3*y(x)^2)*y(x)/x/(x+1), 
       y(x),singsol=all)
 

\[ \frac {x y^{2}}{6 y^{2}+x} = \frac {\left ({\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (x +1\right )^{2} \left ({\mathrm e}^{\textit {\_Z}}+9\right )}{x}\right )+{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+9\right )}+9\right ) x}{54} \]

2.250.4 Mathematica DSolve solution

Solving time : 6.097 (sec)
Leaf size : 75

DSolve[{D[y[x],x] == (y[x]*(x + 3*y[x]^2 + 3*x*y[x]^2))/(x*(1 + x)*(x + 6*y[x]^2)),{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{2 c_1} x}{(x+1)^2}\right )}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{2 c_1} x}{(x+1)^2}\right )}}{\sqrt {6}} \\ y(x)\to 0 \\ \end{align*}