60.2.21 problem 597

Internal problem ID [10608]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 597
Date solved : Monday, January 27, 2025 at 09:17:45 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {2 a}{x^{2} \left (-y+2 F \left (\frac {x y^{2}-4 a}{x}\right ) a \right )} \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 55

dsolve(diff(y(x),x) = 2*a/x^2/(-y(x)+2*F((x*y(x)^2-4*a)/x)*a),y(x), singsol=all)
 

\begin{align*} y &= \operatorname {RootOf}\left (F \left (\frac {x \,\textit {\_Z}^{2}-4 a}{x}\right )\right ) \\ -\frac {y}{2 a}+\frac {\int _{}^{y^{2}-\frac {4 a}{x}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a}}{8 a^{2}}-c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.263 (sec). Leaf size: 130

DSolve[D[y[x],x] == (2*a)/(x^2*(2*a*F[(-4*a + x*y[x]^2)/x] - y[x])),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{2 a F\left (\frac {x K[2]^2-4 a}{x}\right )}-\int _1^x\frac {2 K[2] F''\left (\frac {K[1] K[2]^2-4 a}{K[1]}\right )}{F\left (\frac {K[1] K[2]^2-4 a}{K[1]}\right )^2 K[1]^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {1}{F\left (\frac {K[1] y(x)^2-4 a}{K[1]}\right ) K[1]^2}dK[1]=c_1,y(x)\right ] \]