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: 0.162 (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: 0.208 (sec). Leaf size: 936
DSolve[(y'[x])^2-x*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\left (-2 x^2+\sqrt [3]{8 x^6-20 e^{c_1} x^3+\sqrt {e^{c_1} \left (-8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}\right ){}^2-4 e^{c_1} x}{8 \sqrt [3]{8 x^6-20 e^{c_1} x^3+\sqrt {e^{c_1} \left (-8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}} \\ y(x)\to \frac {1}{16} \left (8 x^2+\frac {\left (4+4 i \sqrt {3}\right ) x \left (x^3+e^{c_1}\right )}{\sqrt [3]{8 x^6-20 e^{c_1} x^3+\sqrt {e^{c_1} \left (-8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}}+\left (1-i \sqrt {3}\right ) \sqrt [3]{8 x^6-20 e^{c_1} x^3+\sqrt {e^{c_1} \left (-8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}\right ) \\ y(x)\to \frac {1}{16} \left (8 x^2+\frac {\left (4-4 i \sqrt {3}\right ) x \left (x^3+e^{c_1}\right )}{\sqrt [3]{8 x^6-20 e^{c_1} x^3+\sqrt {e^{c_1} \left (-8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}}+\left (1+i \sqrt {3}\right ) \sqrt [3]{8 x^6-20 e^{c_1} x^3+\sqrt {e^{c_1} \left (-8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}\right ) \\ y(x)\to \frac {4 e^{c_1} x-\left (-2 x^2+\sqrt [3]{8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}\right ){}^2}{8 \sqrt [3]{8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}} \\ y(x)\to \frac {\left (4+4 i \sqrt {3}\right ) x^4+\left (1-i \sqrt {3}\right ) \left (8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}\right ){}^{2/3}+8 x^2 \sqrt [3]{8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}+\left (-4-4 i \sqrt {3}\right ) e^{c_1} x}{16 \sqrt [3]{8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}} \\ y(x)\to \frac {\left (4-4 i \sqrt {3}\right ) x^4+\left (1+i \sqrt {3}\right ) \left (8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}\right ){}^{2/3}+8 x^2 \sqrt [3]{8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}+4 i \left (\sqrt {3}+i\right ) e^{c_1} x}{16 \sqrt [3]{8 x^6+20 e^{c_1} x^3+\sqrt {e^{c_1} \left (8 x^3+e^{c_1}\right ){}^3}-e^{2 c_1}}} \\ \end{align*}