60.2.205 problem 781

Internal problem ID [10792]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 781
Date solved : Monday, January 27, 2025 at 09:44:53 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {\left (x^{4}+x^{3}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 70

dsolve(diff(y(x),x) = 1/(6*y(x)^2+x)*(x^4+x^3+x+3*y(x)^2)*y(x)/x,y(x), singsol=all)
 

\[ \frac {y^{2} x}{6 y^{2}+x} = \frac {\left ({\mathrm e}^{\operatorname {RootOf}\left (2 x^{3} {\mathrm e}^{\textit {\_Z}}+3 x^{2} {\mathrm e}^{\textit {\_Z}}+3 \,{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-3 \,{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}+9}{x}\right )+9 c_{1} {\mathrm e}^{\textit {\_Z}}+3 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+27\right )}+9\right ) x}{54} \]

✓ Solution by Mathematica

Time used: 0.394 (sec). Leaf size: 89

DSolve[D[y[x],x] == (y[x]*(x + x^3 + x^4 + 3*y[x]^2))/(x*(x + 6*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [-\frac {\int _1^x-\frac {6 y(x)}{K[1]^2}dK[1]{}^2}{2 \int _1^x-\frac {6}{K[1]^2}dK[1]}+\int _1^x\left (-K[1]^2-K[1]-\frac {1}{K[1]}-\frac {3 y(x)^2}{K[1]^2}\right )dK[1]+\frac {3 y(x)^2}{x}+\log (y(x))=c_1,y(x)\right ] \]