60.2.40 problem 616

Internal problem ID [10627]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 616
Date solved : Monday, January 27, 2025 at 09:18:35 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=-\frac {-x^{2}+2 x^{3} y-F \left (\left (y x -1\right ) x \right )}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 38

dsolve(diff(y(x),x) = -1/x^4*(-x^2+2*x^3*y(x)-F((x*y(x)-1)*x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.372 (sec). Leaf size: 177

DSolve[D[y[x],x] == (x^2 + F[x*(-1 + x*y[x])] - 2*x^3*y[x])/x^4,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x^2+F(x (x K[2]-1)) \int _1^x\left (\frac {2 K[2] F''(K[1] (K[1] K[2]-1)) K[1]^3}{F(K[1] (K[1] K[2]-1))^2}-\frac {F''(K[1] (K[1] K[2]-1)) K[1]^2}{F(K[1] (K[1] K[2]-1))^2}-\frac {2 K[1]}{F(K[1] (K[1] K[2]-1))}\right )dK[1]}{F(x (x K[2]-1))}dK[2]+\int _1^x\left (-\frac {2 K[1] y(x)}{F(K[1] (K[1] y(x)-1))}+\frac {1}{F(K[1] (K[1] y(x)-1))}+\frac {1}{K[1]^2}\right )dK[1]=c_1,y(x)\right ] \]