2.339 problem 916

Internal problem ID [9894]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 916
Date solved : Thursday, October 17, 2024 at 10:17:33 PM
CAS classification : [NONE]

Solve

\begin{align*} y^{\prime }&=\frac {y \left (\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )+\ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x \left (x +1\right )} \end{align*}

Unknown ode type.

2.339.1 Maple step by step solution

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

2.339.2 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 
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 = 5`[0, (ln(x)^2+2*ln(x)*ln(y)+ln(y)^2)*y/x]
 

2.339.3 Maple dsolve solution

Solving time : 0.015 (sec)
Leaf size : 73

dsolve(diff(y(x),x) = y(x)*(ln(y(x))*x+ln(y(x))-x-1+x*ln(x)+ln(x)+ln(x)^2*x^4+2*x^4*ln(y(x))*ln(x)+x^4*ln(y(x))^2)/x/(x+1), 
       y(x),singsol=all)
 

\[ y = {\mathrm e}^{\frac {-12 \ln \left (x \right ) \ln \left (x +1\right )+\left (-3 x^{4}+4 x^{3}-6 x^{2}+12 c_{1} +12 x \right ) \ln \left (x \right )-12 x}{3 x^{4}-4 x^{3}+6 x^{2}+12 \ln \left (x +1\right )-12 c_{1} -12 x}} \]

2.339.4 Mathematica DSolve solution

Solving time : 0.577 (sec)
Leaf size : 50

DSolve[{D[y[x],x] == ((-1 - x + Log[x] + x*Log[x] + x^4*Log[x]^2 + Log[y[x]] + x*Log[y[x]] + 2*x^4*Log[x]*Log[y[x]] + x^4*Log[y[x]]^2)*y[x])/(x*(1 + x)),{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\exp \left (\frac {12 x}{-3 x^4+4 x^3-6 x^2+12 x-12 \log (x+1)+c_1}\right )}{x} \\ y(x)\to \frac {1}{x} \\ \end{align*}