17.3 problem Ex 3

Internal problem ID [10213]

Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article 28. Summary. Page 59
Problem number: Ex 3.
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}+1=0} \end {gather*}

Solution by Maple

Time used: 0.047 (sec). Leaf size: 45

dsolve(x^3*diff(y(x),x)^2+x^2*y(x)*diff(y(x),x)+1=0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \frac {x c_{1}^{2}+4}{2 c_{1} x} \\ y \relax (x ) = \frac {c_{1}^{2}+4 x}{2 c_{1} x} \\ y \relax (x ) = \frac {c_{1}}{\sqrt {x}} \\ \end{align*}

Solution by Mathematica

Time used: 1.199 (sec). Leaf size: 215

DSolve[x^3*(y'[x])^2+x^2*y[x]*y'[x]+1==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)^2-4}\right )+\frac {2 \left (\sqrt {y(x)^2}-y(x)\right ) \log \left (\sqrt {x y(x)^2-4}-\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)^2-4}\right )-\frac {2 \left (y(x)+\sqrt {y(x)^2}\right ) \log \left (\sqrt {x y(x)^2-4}-\sqrt {x} \sqrt {y(x)^2}\right )}{y(x)}=c_1,y(x)\right ] \\ y(x)\to -\frac {2}{\sqrt {x}} \\ y(x)\to \frac {2}{\sqrt {x}} \\ \end{align*}