83.6.1 problem 1

Internal problem ID [19117]
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. Exercise II (E) at page 19
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 12:58:22 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.218 (sec). Leaf size: 33

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

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