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 : Monday, January 27, 2025 at 07:52:15 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*}

✓ Solution by Maple

Time used: 0.248 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

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

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

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