69.1.61 problem 80

Internal problem ID [14214]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 80
Date solved : Tuesday, January 28, 2025 at 06:22:19 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.321 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 95

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {x}{x^2+K[2]^2}+K[2]-\int _1^x\left (\frac {2 K[2]^2}{\left (K[1]^2+K[2]^2\right )^2}-\frac {1}{K[1]^2+K[2]^2}\right )dK[1]\right )dK[2]+\int _1^x\left (K[1]-\frac {y(x)}{K[1]^2+y(x)^2}\right )dK[1]=c_1,y(x)\right ] \]