60.2.19 problem 595

Internal problem ID [10606]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 595
Date solved : Tuesday, January 28, 2025 at 04:55:52 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.267 (sec). Leaf size: 92

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

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

✓ Solution by Mathematica

Time used: 0.211 (sec). Leaf size: 204

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

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