Internal problem ID [6042]
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: 3.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+y^{\prime } x^{3}-2 y x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.393 (sec). Leaf size: 23
dsolve(diff(y(x),x)^2+x^3*diff(y(x),x)-2*x^2*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x^{4}}{8} \\ y \relax (x ) = c_{1} x^{2}+2 c_{1}^{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.838 (sec). Leaf size: 142
DSolve[(y'[x])^2+x^3*y'[x]-2*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{4} \log (y(x))-\frac {\sqrt {x^6+8 x^2 y(x)} \tanh ^{-1}\left (\frac {\sqrt {x^4+8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4+8 y(x)}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6+8 x^2 y(x)} \tanh ^{-1}\left (\frac {\sqrt {x^4+8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4+8 y(x)}}+\frac {1}{4} \log (y(x))=c_1,y(x)\right ] \\ y(x)\to -\frac {x^4}{8} \\ \end{align*}