60.2.6 problem 582

Internal problem ID [10593]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 582
Date solved : Tuesday, January 28, 2025 at 04:55:20 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 48

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

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

✓ Solution by Mathematica

Time used: 0.279 (sec). Leaf size: 142

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

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