60.2.282 problem 858

Internal problem ID [10869]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 858
Date solved : Monday, January 27, 2025 at 10:18:59 PM
CAS classification : [[_homogeneous, `class C`], _Abel]

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

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 42

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

\[ y = \frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{3} a +a \,\textit {\_a}^{2}+a +b}d \textit {\_a} \right ) a -x +c_{1} \right ) a -b x}{a} \]

✓ Solution by Mathematica

Time used: 0.222 (sec). Leaf size: 92

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

\[ \text {Solve}\left [\int _1^{\frac {\frac {a+3 b x}{a}+3 y(x)}{\sqrt [3]{\frac {29 a+27 b}{a}}}}\frac {1}{K[1]^3-\frac {3 a^{2/3} K[1]}{(29 a+27 b)^{2/3}}+1}dK[1]=\frac {1}{9} x \left (\frac {29 a+27 b}{a}\right )^{2/3}+c_1,y(x)\right ] \]