28.1.52 problem 53

Internal problem ID [4358]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 53
Date solved : Monday, January 27, 2025 at 09:08:54 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 2 x^{3} y+y^{3}-\left (x^{4}+2 x y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 1.453 (sec). Leaf size: 149

dsolve((2*x^3*y(x)+y(x)^3)-( x^4+2*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-x^{{3}/{2}} \operatorname {RootOf}\left (-16+x^{7} c_{1} \textit {\_Z}^{12}-4 c_{1} x^{{11}/{2}} \textit {\_Z}^{10}+6 c_{1} x^{4} \textit {\_Z}^{8}+\left (128 x^{{9}/{2}}-4 c_{1} x^{{5}/{2}}\right ) \textit {\_Z}^{6}+\left (-192 x^{3}+c_{1} x \right ) \textit {\_Z}^{4}+96 x^{{3}/{2}} \textit {\_Z}^{2}\right )^{2}+1}{2 \operatorname {RootOf}\left (-16+x^{7} c_{1} \textit {\_Z}^{12}-4 c_{1} x^{{11}/{2}} \textit {\_Z}^{10}+6 c_{1} x^{4} \textit {\_Z}^{8}+\left (128 x^{{9}/{2}}-4 c_{1} x^{{5}/{2}}\right ) \textit {\_Z}^{6}+\left (-192 x^{3}+c_{1} x \right ) \textit {\_Z}^{4}+96 x^{{3}/{2}} \textit {\_Z}^{2}\right )^{2}} \]

✓ Solution by Mathematica

Time used: 60.168 (sec). Leaf size: 2023

DSolve[(2*x^3*y[x]+y[x]^3)-( x^4+2*x*y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {48 x^3+\frac {e^{4 c_1} x^2}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}+\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}+e^{2 c_1} \left (-x-\frac {96 x^4}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}\right )}}{8 \sqrt {3}} \\ y(x)\to \frac {\sqrt {48 x^3+\frac {e^{4 c_1} x^2}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}+\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}+e^{2 c_1} \left (-x-\frac {96 x^4}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}\right )}}{8 \sqrt {3}} \\ y(x)\to -\frac {\sqrt {\frac {i \left (\sqrt {3}+i\right ) e^{4 c_1} x^2+96 x^3 \sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}-2 e^{2 c_1} x \left (48 i \left (\sqrt {3}+i\right ) x^3+\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}\right )+\left (-1-i \sqrt {3}\right ) \left (-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}\right ){}^{2/3}}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}}}{8 \sqrt {6}} \\ y(x)\to \frac {\sqrt {\frac {i \left (\sqrt {3}+i\right ) e^{4 c_1} x^2+96 x^3 \sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}-2 e^{2 c_1} x \left (48 i \left (\sqrt {3}+i\right ) x^3+\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}\right )+\left (-1-i \sqrt {3}\right ) \left (-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}\right ){}^{2/3}}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}}}{8 \sqrt {6}} \\ y(x)\to -\frac {\sqrt {\frac {-i \left (\sqrt {3}-i\right ) e^{4 c_1} x^2+96 x^3 \sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}+i \left (\sqrt {3}+i\right ) \left (-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}\right ){}^{2/3}+e^{2 c_1} \left (96 \left (1+i \sqrt {3}\right ) x^4-2 x \sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}\right )}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}}}{8 \sqrt {6}} \\ y(x)\to \frac {\sqrt {\frac {-i \left (\sqrt {3}-i\right ) e^{4 c_1} x^2+96 x^3 \sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}+i \left (\sqrt {3}+i\right ) \left (-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}\right ){}^{2/3}+e^{2 c_1} \left (96 \left (1+i \sqrt {3}\right ) x^4-2 x \sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}\right )}{\sqrt [3]{-3456 e^{2 c_1} x^7+144 e^{4 c_1} x^5-e^{6 c_1} x^3+192 \sqrt {3} \sqrt {-e^{4 c_1} x^{12} \left (-108 x^2+e^{2 c_1}\right )}}}}}{8 \sqrt {6}} \\ \end{align*}