Internal problem ID [2688]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {2 y x^{3}+y^{3}-\left (x^{4}+2 x y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.041 (sec). Leaf size: 49
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)
\[ \ln \relax (x )-c_{1}+\frac {3 \ln \left (-\frac {x^{\frac {3}{2}}-2 y \relax (x )}{x^{\frac {3}{2}}}\right )}{2}-2 \ln \left (\frac {y \relax (x )}{x^{\frac {3}{2}}}\right )+\frac {3 \ln \left (\frac {x^{\frac {3}{2}}+2 y \relax (x )}{x^{\frac {3}{2}}}\right )}{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.146 (sec). Leaf size: 2019
DSolve[(2*x^3*y[x]+y[x]^3)-( x^4+2*x*y[x]^2)*y'[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 )}}}+e^{2 c_1} x \left (-1-\frac {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 )}}}\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 {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 )}}}+e^{2 c_1} x \left (-1-\frac {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 )}}}\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 {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 {\left (-1-i \sqrt {3}\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 {\left (-1-i \sqrt {3}\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*}