Internal problem ID [3416]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 677.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (x^{3}+y^{3}\right ) y^{\prime }+x^{2} \left (a x +3 y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 29
dsolve((x^3+y(x)^3)*diff(y(x),x)+x^2*(a*x+3*y(x)) = 0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\RootOf \left (a \,x^{4} c_{1}^{\frac {4}{3}}+4 x^{3} c_{1} \textit {\_Z} +\textit {\_Z}^{4}-1\right )}{c_{1}^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.158 (sec). Leaf size: 1402
DSolve[(x^3+y[x]^3)y'[x]+x^2(a x+3 y[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}-\sqrt {-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}+\frac {-a x^4+e^{4 c_1}}{\sqrt [3]{3 x^6+\sqrt {9 x^{12}+\frac {1}{3} \left (-a x^4+e^{4 c_1}\right ){}^3}}}-\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}}}}{\sqrt {2} \sqrt [3]{3}} \\ y(x)\to \frac {\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}+\sqrt {-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}+\frac {-a x^4+e^{4 c_1}}{\sqrt [3]{3 x^6+\sqrt {9 x^{12}+\frac {1}{3} \left (-a x^4+e^{4 c_1}\right ){}^3}}}-\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}}}}{\sqrt {2} \sqrt [3]{3}} \\ y(x)\to -\frac {\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}+\sqrt {-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}+\frac {-a x^4+e^{4 c_1}}{\sqrt [3]{3 x^6+\sqrt {9 x^{12}+\frac {1}{3} \left (-a x^4+e^{4 c_1}\right ){}^3}}}+\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}}}}{\sqrt {2} \sqrt [3]{3}} \\ y(x)\to \frac {\sqrt {-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}+\frac {-a x^4+e^{4 c_1}}{\sqrt [3]{3 x^6+\sqrt {9 x^{12}+\frac {1}{3} \left (-a x^4+e^{4 c_1}\right ){}^3}}}+\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}}}-\sqrt {\frac {\sqrt [3]{3} a x^4+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}\right ){}^{2/3}-\sqrt [3]{3} e^{4 c_1}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (-a x^4+e^{4 c_1}\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}} \\ \end{align*}