60.2.328 problem 905

Internal problem ID [10915]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 905
Date solved : Tuesday, January 28, 2025 at 05:30:51 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

\begin{align*} y^{\prime }&=\frac {a^{2} x +a^{3} x^{3}+a^{3} x^{3} y^{2}+2 a^{2} x^{2} y+a x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1}{a^{3} x^{3}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 46

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

\[ y = \frac {29 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+x +3 c_{1} \right ) a x -3 a x -9}{9 a x} \]

✓ Solution by Mathematica

Time used: 0.191 (sec). Leaf size: 63

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

\[ \text {Solve}\left [\int _1^{\frac {\frac {a x+3}{a x}+3 y(x)}{\sqrt [3]{29}}}\frac {1}{K[1]^3-\frac {3 K[1]}{29^{2/3}}+1}dK[1]=\frac {1}{9} 29^{2/3} x+c_1,y(x)\right ] \]