2.268 problem 844

Internal problem ID [9828]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 844
Date solved : Friday, October 11, 2024 at 11:59:15 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`]]

Solve

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

Unknown ode type.

2.268.1 Maple step by step solution

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

2.268.2 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 
trying Abel 
<- Abel successful`
 

2.268.3 Maple dsolve solution

Solving time : 0.003 (sec)
Leaf size : 106

dsolve(diff(y(x),x) = y(x)*(x+y(x))*(y(x)+1)/x/(y(x)*x+x+y(x)), 
       y(x),singsol=all)
 

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

2.268.4 Mathematica DSolve solution

Solving time : 9.288 (sec)
Leaf size : 386

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

\[ \text {Solve}\left [\frac {2^{2/3} \left (1-\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}\right ) \left (\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}+2\right ) \left (\left (1-\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}\right ) \log \left (2^{2/3} \left (1-\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}\right )\right )+\left (\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}-1\right ) \log \left (2^{2/3} \left (\frac {\left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}+2\right )\right )-3\right )}{9 \left (\frac {3 \left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2 ((x-2) y(x)+x)}{x^4 ((x+1) y(x)+x)}-\frac {((x-2) y(x)+x)^3}{((x+1) y(x)+x)^3}-2\right )}=\frac {2^{2/3} \left (\frac {x^6}{(x+1)^3}\right )^{2/3} (x+1)^2}{9 x^3}+c_1,y(x)\right ] \]