29.25.10 problem 707

Internal problem ID [5297]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 707
Date solved : Monday, January 27, 2025 at 11:04:55 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (x^{3}-y^{4}\right ) y^{\prime }&=3 x^{2} y \end{align*}

✓ Solution by Maple

Time used: 0.127 (sec). Leaf size: 25

dsolve((x^3-y(x)^4)*diff(y(x),x) = 3*x^2*y(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \operatorname {RootOf}\left (x^{9} \textit {\_Z}^{4}+3-{\mathrm e}^{\frac {9 c_{1}}{4}} \textit {\_Z} \right ) x^{3} \]

✓ Solution by Mathematica

Time used: 60.135 (sec). Leaf size: 1021

DSolve[(x^3-y[x]^4)D[y[x],x]==3 x^2 y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}} \\ y(x)\to \frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}}-\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}} \\ \end{align*}