Internal problem ID [6801]
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]
\[ \boxed {x^{4} {y^{\prime }}^{2}+2 y y^{\prime } x^{3}=4} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 49
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 \left (x \right ) &= -\frac {2 i}{x} \\ y \left (x \right ) &= \frac {2 i}{x} \\ y \left (x \right ) &= \frac {2 \sinh \left (-\ln \left (x \right )+c_{1} \right )}{x} \\ y \left (x \right ) &= -\frac {2 \sinh \left (-\ln \left (x \right )+c_{1} \right )}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.688 (sec). Leaf size: 71
DSolve[x^4*(y'[x])^2+2*x^3*y[x]*y'[x]-4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {4 e^{c_1}}{x^2}-\frac {e^{-c_1}}{4} \\ y(x)\to \frac {e^{-c_1}}{4}-\frac {4 e^{c_1}}{x^2} \\ y(x)\to -\frac {2 i}{x} \\ y(x)\to \frac {2 i}{x} \\ \end{align*}