Internal problem ID [6814]
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: 23.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, _dAlembert]
\[ \boxed {4 x {y^{\prime }}^{2}-3 y y^{\prime }=-3} \]
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 123
dsolve(4*x*diff(y(x),x)^2-3*y(x)*diff(y(x),x)+3=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2 x \left (6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}} \\ y \left (x \right ) &= \frac {2 x \left (6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}} \\ y \left (x \right ) &= \frac {2 x \left (-6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}} \\ y \left (x \right ) &= -\frac {2 x \left (-6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.695 (sec). Leaf size: 187
DSolve[4*x*(y'[x])^2-3*y[x]*y'[x]+3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {432 x-e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}} \\ y(x)\to \frac {\sqrt {432 x-e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}} \\ y(x)\to -\frac {\sqrt {432 x+e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}} \\ y(x)\to \frac {\sqrt {432 x+e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}} \\ \end{align*}