60.2.29 problem 605

Internal problem ID [10616]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 605
Date solved : Monday, January 27, 2025 at 09:18:06 PM
CAS classification : [NONE]

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

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 43

dsolve(diff(y(x),x) = -1/4*y(x)^2*(2*x-F(-1/2*(-2+x*y(x))/y(x)))/x,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.757 (sec). Leaf size: 145

DSolve[D[y[x],x] == -1/4*((2*x - F[(1 - (x*y[x])/2)/y[x]])*y[x]^2)/x,y[x],x,IncludeSingularSolutions -> True]
 

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