Internal problem ID [6048]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 99. Clairaut’s equation. EXERCISES Page
320
Problem number: 9.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {x^{4} \left (y^{\prime }\right )^{2}+2 x^{3} y y^{\prime }-4=0} \end {gather*}
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 39
dsolve(x^4*diff(y(x),x)^2+2*x^3*y(x)*diff(y(x),x)-4=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {2 \sinh \left (-\ln \relax (x )+c_{1}\right )}{x} \\ y \relax (x ) = -\frac {2 \sinh \left (-\ln \relax (x )+c_{1}\right )}{x} \\ y \relax (x ) = \frac {c_{1}}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.815 (sec). Leaf size: 343
DSolve[x^4*(y'[x])^2+2*x^3*y[x]*y'[x]-4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\left (-y(x)-\sqrt {y(x)^2}\right ) \log \left (\sqrt {x^6 y(x)^2+4 x^4}-x^3 \sqrt {y(x)^2}\right )}{y(x)}+\log \left (x^3 \left (-\sqrt {y(x)^2}\right )-2 x^2+\sqrt {x^6 y(x)^2+4 x^4}\right )+\log \left (x^3 \left (-\sqrt {y(x)^2}\right )+2 x^2+\sqrt {x^6 y(x)^2+4 x^4}\right )-\log (y(x))+\frac {2 \left (\sqrt {y(x)^2}-y(x)\right ) \log (x)}{y(x)}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\left (\sqrt {y(x)^2}-y(x)\right ) \log \left (\sqrt {x^6 y(x)^2+4 x^4}-x^3 \sqrt {y(x)^2}\right )}{y(x)}+\log \left (x^3 \left (-\sqrt {y(x)^2}\right )-2 x^2+\sqrt {x^6 y(x)^2+4 x^4}\right )+\log \left (x^3 \left (-\sqrt {y(x)^2}\right )+2 x^2+\sqrt {x^6 y(x)^2+4 x^4}\right )-\log (y(x))-\frac {2 \left (y(x)+\sqrt {y(x)^2}\right ) \log (x)}{y(x)}=c_1,y(x)\right ] \\ y(x)\to -\frac {2 i}{x} \\ y(x)\to \frac {2 i}{x} \\ \end{align*}