60.2.28 problem 604

Internal problem ID [10615]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 604
Date solved : Tuesday, January 28, 2025 at 04:56:04 PM
CAS classification : [`x=_G(y,y')`]

\begin{align*} y^{\prime }&=\frac {2 y^{3}}{1+2 F \left (\frac {1+4 x y^{2}}{y^{2}}\right ) y} \end{align*}

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 47

dsolve(diff(y(x),x) = 2*y(x)^3/(1+2*F((1+4*x*y(x)^2)/y(x)^2)*y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.325 (sec). Leaf size: 143

DSolve[D[y[x],x] == (2*y[x]^3)/(1 + 2*F[(1 + 4*x*y[x]^2)/y[x]^2]*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x\frac {\left (\frac {8 K[1]}{K[2]}-\frac {2 \left (4 K[1] K[2]^2+1\right )}{K[2]^3}\right ) F''\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )}{F\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )^2}dK[1]+\frac {1}{K[2]^2}+\frac {1}{2 F\left (\frac {4 x K[2]^2+1}{K[2]^2}\right ) K[2]^3}\right )dK[2]+\int _1^x-\frac {1}{F\left (\frac {4 K[1] y(x)^2+1}{y(x)^2}\right )}dK[1]=c_1,y(x)\right ] \]