83.8.1 problem 1

Internal problem ID [19127]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 12:59:00 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.095 (sec). Leaf size: 152

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

\[ -\frac {\arctan \left (\frac {a^{2}-2 x^{2}-2 y \left (x \right )^{2}}{2 \sqrt {\left (x^{2}+y \left (x \right )^{2}\right ) \left (a^{2}-x^{2}-y \left (x \right )^{2}\right )}}\right ) \left (x^{2}+y \left (x \right )^{2}\right ) \sqrt {\frac {a^{2}-x^{2}-y \left (x \right )^{2}}{x^{2}+y \left (x \right )^{2}}}+2 \sqrt {\left (x^{2}+y \left (x \right )^{2}\right ) \left (a^{2}-x^{2}-y \left (x \right )^{2}\right )}\, \left (c_{1} -\arctan \left (\frac {x}{y \left (x \right )}\right )\right )}{2 \sqrt {\left (x^{2}+y \left (x \right )^{2}\right ) \left (a^{2}-x^{2}-y \left (x \right )^{2}\right )}} = 0 \]

✓ Solution by Mathematica

Time used: 10.890 (sec). Leaf size: 116

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

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