12.5.50 problem 49

Internal problem ID [1674]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 49
Date solved : Saturday, March 29, 2025 at 11:26:32 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`], _Riccati]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

ode:=x*ln(x)^2*diff(y(x),x) = -4*ln(x)^2+ln(x)*y(x)+y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 i \tan \left (2 i \ln \left (\ln \left (x \right )\right )+c_1 \right ) \ln \left (x \right ) \]

✓ Mathematica. Time used: 1.021 (sec). Leaf size: 64

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

\begin{align*} y(x)\to 2 i \log (x) \tan (2 i \log (\log (x))+c_1) \\ y(x)\to \frac {2 \log (x) \left (-\log ^4(x)+e^{2 i \text {Interval}[\{0,\pi \}]}\right )}{\log ^4(x)+e^{2 i \text {Interval}[\{0,\pi \}]}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*log(x)**2*Derivative(y(x), x) - y(x)**2 - y(x)*log(x) + 4*log(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - y(x)**2/(x*log(x)**2) - y(x)/(x*log(x)) + 4/x cannot be solved by the factorable group method