29.24.15 problem 677

Internal problem ID [5268]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 677
Date solved : Monday, January 27, 2025 at 10:51:02 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \left (x^{3}+y^{3}\right ) y^{\prime }+x^{2} \left (a x +3 y\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.060 (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)
 

\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (a \,x^{4} c_{1}^{{4}/{3}}+4 x^{3} c_{1} \textit {\_Z} +\textit {\_Z}^{4}-1\right )}{c_{1}^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 60.157 (sec). Leaf size: 1430

DSolve[(x^3+y[x]^3)D[y[x],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 {\sqrt [3]{3} \left (-a x^4+e^{4 c_1}\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\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 {\sqrt [3]{3} \left (-a x^4+e^{4 c_1}\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\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 {\sqrt [3]{3} \left (-a x^4+e^{4 c_1}\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\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 {\sqrt [3]{3} \left (-a x^4+e^{4 c_1}\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\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*}