60.1.364 problem 365

Internal problem ID [10378]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 365
Date solved : Monday, January 27, 2025 at 07:38:21 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.135 (sec). Leaf size: 34

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

\[ y = \cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} -\int _{}^{\csc \left (\textit {\_Z} \right )^{2} x^{2}}\frac {f \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} +2 c_{1} \right )\right ) x \]

✓ Solution by Mathematica

Time used: 0.252 (sec). Leaf size: 156

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

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