20.4.21 problem Problem 29(b)

Internal problem ID [3656]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 29(b)
Date solved : Sunday, March 30, 2025 at 02:01:44 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \end{align*}

✓ Maple. Time used: 0.188 (sec). Leaf size: 30

ode:=diff(y(x),x) = (x+1/2*y(x))/(1/2*x-y(x)); 
ic:=y(1) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \tan \left (\operatorname {RootOf}\left (4 \textit {\_Z} -4 \ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )-8 \ln \left (x \right )+4 \ln \left (2\right )-\pi \right )\right ) x \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 42

ode=D[y[x],x]==(x+1/2*y[x])/(1/2*x-y[x]); 
ic={y[1]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.038 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x + y(x)/2)/(x/2 - y(x)),0) 
ics = {y(1): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \log {\left (x \right )} = - \log {\left (\sqrt {1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} \right )} + \frac {\operatorname {atan}{\left (\frac {y{\left (x \right )}}{x} \right )}}{2} - \frac {\pi }{8} + \frac {\log {\left (2 \right )}}{2} \]