60.2.33 problem 609

Internal problem ID [10620]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 609
Date solved : Tuesday, January 28, 2025 at 04:56:13 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 22

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

\[ y = \frac {\operatorname {RootOf}\left (x -\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.249 (sec). Leaf size: 117

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

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x^3+F\left (x^3 K[2]\right ) \int _1^x\left (\frac {3 K[1]^5 K[2] F''\left (K[1]^3 K[2]\right )}{F\left (K[1]^3 K[2]\right )^2}-\frac {3 K[1]^2}{F\left (K[1]^3 K[2]\right )}\right )dK[1]}{F\left (x^3 K[2]\right )}dK[2]+\int _1^x\left (1-\frac {3 K[1]^2 y(x)}{F\left (K[1]^3 y(x)\right )}\right )dK[1]=c_1,y(x)\right ] \]