53.3.20 problem 23

Internal problem ID [8482]
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 equation. EXERCISES Page 320
Problem number : 23
Date solved : Monday, January 27, 2025 at 04:07:09 PM
CAS classification : [[_homogeneous, `class G`], _rational, _dAlembert]

\begin{align*} 4 x {y^{\prime }}^{2}-3 y^{\prime } y+3&=0 \end{align*}

✓ Solution by Maple

Time used: 0.033 (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 &= -\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 &= \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 &= \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 &= -\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: 25.891 (sec). Leaf size: 187

DSolve[4*x*(D[y[x],x])^2-3*y[x]*D[y[x],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*}