60.2.31 problem 607

Internal problem ID [10618]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 607
Date solved : Tuesday, January 28, 2025 at 04:56:10 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 0.206 (sec). Leaf size: 121

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

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