28.1.76 problem 79

Internal problem ID [4382]
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 : 79
Date solved : Monday, January 27, 2025 at 09:09:41 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 6 y^{2}-x \left (2 x^{3}+y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.713 (sec). Leaf size: 193

dsolve(6*y(x)^2-(x*(2*x^3+y(x)))*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {x^{3} \left (-x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}-4 c_{1} \right )}{2 c_{1}} \\ y \left (x \right ) &= \frac {x^{3} \left (x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}+4 c_{1} \right )}{2 c_{1}} \\ y \left (x \right ) &= -\frac {x^{3} \left (-x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}-4 c_{1} \right )}{2 c_{1}} \\ y \left (x \right ) &= \frac {x^{3} \left (x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}+4 c_{1} \right )}{2 c_{1}} \\ y \left (x \right ) &= -\frac {x^{3} \left (-x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}-4 c_{1} \right )}{2 c_{1}} \\ y \left (x \right ) &= \frac {x^{3} \left (x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}+4 c_{1} \right )}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.307 (sec). Leaf size: 123

DSolve[6*y[x]^2-(x*(2*x^3+y[x]))*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 2 x^3 \left (-1+\frac {2}{1-\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right ) \\ y(x)\to 2 x^3 \left (-1+\frac {2}{1+\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right ) \\ y(x)\to 0 \\ y(x)\to 2 x^3 \\ y(x)\to \frac {2 \left (\left (x^3\right )^{3/2}-x^{9/2}\right )}{x^{3/2}+\sqrt {x^3}} \\ \end{align*}