60.2.219 problem 795

Internal problem ID [10806]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 795
Date solved : Monday, January 27, 2025 at 10:01:09 PM
CAS classification : [[_homogeneous, `class C`], _rational, _Abel]

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 0.394 (sec). Leaf size: 92

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

\[ \text {Solve}\left [\int _1^{\frac {\frac {3 y(x)}{(a+x)^3}+\frac {1}{(a+x)^2}}{\sqrt [3]{38} \sqrt [3]{\frac {1}{(a+x)^6}}}}\frac {1}{K[1]^3-\frac {6 \sqrt [3]{2} K[1]}{19^{2/3}}+1}dK[1]=\frac {1}{9} 38^{2/3} \left (\frac {1}{(a+x)^6}\right )^{2/3} (a+x)^4 \log (a+x)+c_1,y(x)\right ] \]