6.4.8 problem 8

Internal problem ID [1615]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Existence and Uniqueness of Solutions of Nonlinear Equations. Section 2.3 Page 60
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 04:40:08 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }&=\frac {2 x +3 y}{x -4 y} \end{align*}

✓ Maple. Time used: 0.143 (sec). Leaf size: 51

ode:=diff(y(x),x) = (2*x+3*y(x))/(x-4*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x \left (\sqrt {7}\, \tan \left (\operatorname {RootOf}\left (\sqrt {7}\, \ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+\sqrt {7}\, \ln \left (7\right )-3 \sqrt {7}\, \ln \left (2\right )+2 \sqrt {7}\, c_1 -4 \textit {\_Z} \right )\right )-1\right )}{4} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 53

ode=D[y[x],x]==(2*x+3*y[x])/(x-4*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\log \left (\frac {2 y(x)^2}{x^2}+\frac {y(x)}{x}+1\right )-\frac {4 \arctan \left (\frac {\frac {4 y(x)}{x}+1}{\sqrt {7}}\right )}{\sqrt {7}}=-2 \log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.806 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*x + 3*y(x))/(x - 4*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} - \log {\left (\sqrt {\frac {x^{2}}{y^{2}{\left (x \right )}} + \frac {x}{y{\left (x \right )}} + 2} \right )} - \frac {2 \sqrt {7} \operatorname {atan}{\left (\frac {\sqrt {7} \left (\frac {2 x}{y{\left (x \right )}} + 1\right )}{7} \right )}}{7} \]