60.2.348 problem 925

Internal problem ID [10935]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 925
Date solved : Monday, January 27, 2025 at 10:25:38 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.246 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 9.581 (sec). Leaf size: 228

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

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