Internal problem ID [6061]
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]
Solve \begin {gather*} \boxed {4 x \left (y^{\prime }\right )^{2}-3 y y^{\prime }+3=0} \end {gather*}
✓ Solution by Maple
Time used: 0.153 (sec). Leaf size: 153
dsolve(4*x*diff(y(x),x)^2-3*y(x)*diff(y(x),x)+3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {2 \sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}}{3}-\frac {2 x}{\sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}} \\ y \relax (x ) = \frac {2 \sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}}{3}+\frac {2 x}{\sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}} \\ y \relax (x ) = -\frac {2 \sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}}{3}-\frac {2 x}{\sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}} \\ y \relax (x ) = \frac {2 \sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}}{3}+\frac {2 x}{\sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 11.077 (sec). Leaf size: 163
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-\frac {(-144 x+c_1){}^{3/2}}{\sqrt {c_1}}+c_1}}{6 \sqrt {3}} \\ y(x)\to \frac {\sqrt {432 x-\frac {(-144 x+c_1){}^{3/2}}{\sqrt {c_1}}+c_1}}{6 \sqrt {3}} \\ y(x)\to -\frac {\sqrt {432 x+\frac {(-144 x+c_1){}^{3/2}}{\sqrt {c_1}}+c_1}}{6 \sqrt {3}} \\ y(x)\to \frac {\sqrt {432 x+\frac {(-144 x+c_1){}^{3/2}}{\sqrt {c_1}}+c_1}}{6 \sqrt {3}} \\ \end{align*}