Internal problem ID [6035]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 97. The p-discriminant equation. EXERCISES
Page 314
Problem number: 10.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.312 (sec). Leaf size: 77
dsolve(diff(y(x),x)^2-x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} \frac {c_{1}}{\sqrt {2 x -2 \sqrt {x^{2}+4 y \relax (x )}}}+\frac {2 x}{3}+\frac {\sqrt {x^{2}+4 y \relax (x )}}{3} = 0 \\ \frac {c_{1}}{\sqrt {2 x +2 \sqrt {x^{2}+4 y \relax (x )}}}+\frac {2 x}{3}-\frac {\sqrt {x^{2}+4 y \relax (x )}}{3} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.277 (sec). Leaf size: 965
DSolve[(y'[x])^2-x*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (x^2+\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ){}^2+8 e^{3 c_1} x}{4 \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}} \\ y(x)\to \frac {1}{8} \left (4 x^2+\frac {\left (-1-i \sqrt {3}\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{8} \left (4 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) \\ y(x)\to \frac {2 \sqrt [3]{2} x^4+2^{2/3} \left (-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}\right ){}^{2/3}+4 x^2 \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}-4 \sqrt [3]{2} e^{3 c_1} x}{8 \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}} \\ y(x)\to \frac {1}{16} \left (8 x^2-\frac {4 \sqrt [3]{-2} x \left (x^3-2 e^{3 c_1}\right )}{\sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}}+2 (-2)^{2/3} \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}\right ) \\ y(x)\to \frac {x^2}{2}+\frac {(-1)^{2/3} x \left (x^3-2 e^{3 c_1}\right )}{2\ 2^{2/3} \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}}-\frac {1}{4} \sqrt [3]{-\frac {1}{2}} \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}} \\ \end{align*}