Internal problem ID [8375]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 795.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _rational, _Abel]
Solve \begin {gather*} \boxed {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}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (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)
\[ y \relax (x ) = -\RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{3}-\textit {\_a}^{2}-\textit {\_a} -1}d \textit {\_a} \right )+\ln \left (x +a \right )+c_{1}\right ) \left (x +a \right ) \]
✓ Solution by Mathematica
Time used: 0.3 (sec). Leaf size: 111
DSolve[y'[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]
\[ \text {Solve}\left [-\frac {19}{3} \text {RootSum}\left [-19 \text {$\#$1}^3+6 \sqrt [3]{38} \text {$\#$1}-19\&,\frac {\log \left (\frac {\frac {3 y(x)}{(a+x)^3}+\frac {1}{(a+x)^2}}{\sqrt [3]{38} \sqrt [3]{\frac {1}{(a+x)^6}}}-\text {$\#$1}\right )}{2 \sqrt [3]{38}-19 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 38^{2/3} \left (\frac {1}{(a+x)^6}\right )^{2/3} (a+x)^4 \log (a+x)+c_1,y(x)\right ] \]