60.2.44 problem 620

Internal problem ID [10631]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 620
Date solved : Monday, January 27, 2025 at 09:18:48 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.227 (sec). Leaf size: 37

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} +\int _{}^{{\mathrm e}^{2 \textit {\_Z}}-2 x \,{\mathrm e}^{\textit {\_Z}}}\frac {1}{{\mathrm e}^{2 F \left (\textit {\_a} \right )}+\textit {\_a}}d \textit {\_a} +c_{1} \right )}-x \]

✓ Solution by Mathematica

Time used: 0.716 (sec). Leaf size: 205

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

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