60.2.20 problem 596

Internal problem ID [10607]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 596
Date solved : Monday, January 27, 2025 at 09:17:43 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.246 (sec). Leaf size: 156

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

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