60.2.23 problem 599

Internal problem ID [10610]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 599
Date solved : Tuesday, January 28, 2025 at 04:55:54 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 69

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

\begin{align*} y &= \operatorname {RootOf}\left (F \left (\textit {\_Z}^{2}+x^{2}\right )\right ) \\ y &= \sqrt {-x^{2}+\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +2 c_{1} \right )} \\ y &= -\sqrt {-x^{2}+\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +2 c_{1} \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.166 (sec). Leaf size: 95

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

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