60.2.25 problem 601

Internal problem ID [10612]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 601
Date solved : Tuesday, January 28, 2025 at 04:55:57 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 77

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

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

✓ Solution by Mathematica

Time used: 0.218 (sec). Leaf size: 182

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

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