problem 923

Internal problem ID [10933]
Book : Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear ﬁrst order
Problem number : 923
Date solved : Monday, January 27, 2025 at 10:25:32 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 ) \left (x +y\right )}}{y^{2}+2 y x +x^{2}-{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}} \end{align*}

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

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

60.2.346 problem 923