29.23.13 problem 644

Internal problem ID [5235]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 644
Date solved : Monday, January 27, 2025 at 10:37:34 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} \left (x \left (a -x^{2}-y^{2}\right )+y\right ) y^{\prime }+x -\left (a -x^{2}-y^{2}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.588 (sec). Leaf size: 37

dsolve((x*(a-x^2-y(x)^2)+y(x))*diff(y(x),x)+x-(a-x^2-y(x)^2)*y(x) = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \cot \left (\operatorname {RootOf}\left (2 c_{1} a -2 \textit {\_Z} a +\ln \left (-\frac {x^{2}}{a \sin \left (\textit {\_Z} \right )^{2}-x^{2}}\right )\right )\right ) x \]

✓ Solution by Mathematica

Time used: 0.188 (sec). Leaf size: 47

DSolve[(x*(a-x^2-y[x]^2)+y[x])*D[y[x],x]+x-(a-x^2-y[x]^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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