2.1 problem 1

Internal problem ID [6112]

Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 1.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [[_homogeneous, class G]]

Solve \begin {gather*} \boxed {x^{3} \left (y^{\prime }\right )^{2}+y^{\prime } y x^{2}+4=0} \end {gather*}

Solution by Maple

Time used: 0.484 (sec). Leaf size: 45

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

\begin{align*} y \relax (x ) = \frac {c_{1}^{2} x +16}{2 c_{1} x} \\ y \relax (x ) = \frac {c_{1}^{2}+16 x}{2 c_{1} x} \\ y \relax (x ) = \frac {c_{1}}{\sqrt {x}} \\ \end{align*}

Solution by Mathematica

Time used: 1.862 (sec). Leaf size: 215

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

\begin{align*} \text {Solve}\left [-2 \log (y(x))+2 \log \left (x y(x)^2-\sqrt {x} \sqrt {y(x)^2} \sqrt {x y(x)^2-16}\right )+\frac {2 \left (\sqrt {y(x)^2}-y(x)\right ) \log \left (\sqrt {x y(x)^2-16}-\sqrt {x} \sqrt {y(x)^2}\right )}{y(x)}=c_1,y(x)\right ] \\ \text {Solve}\left [-2 \log (y(x))+2 \log \left (x y(x)^2-\sqrt {x} \sqrt {y(x)^2} \sqrt {x y(x)^2-16}\right )-\frac {2 \left (y(x)+\sqrt {y(x)^2}\right ) \log \left (\sqrt {x y(x)^2-16}-\sqrt {x} \sqrt {y(x)^2}\right )}{y(x)}=c_1,y(x)\right ] \\ y(x)\to -\frac {4}{\sqrt {x}} \\ y(x)\to \frac {4}{\sqrt {x}} \\ \end{align*}