60.2.276 problem 852

Internal problem ID [10863]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 852
Date solved : Monday, January 27, 2025 at 10:17:58 PM
CAS classification : [[_homogeneous, `class C`], _Abel]

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

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 0.208 (sec). Leaf size: 92

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

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