Internal problem ID [6092]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES
Page 324
Problem number: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 21
dsolve(x^2*diff(y(x),x$2)+diff(y(x),x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {x}{c_{1}}+\frac {\ln \left (c_{1} x -1\right )}{c_{1}^{2}}+c_{2} \]
✓ Solution by Mathematica
Time used: 0.94 (sec). Leaf size: 47
DSolve[x^2*y''[x]+(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x}{c_1}+\frac {\log (1+c_1 x)}{c_1{}^2}+c_2 \\ y(x)\to c_2 \\ y(x)\to -\frac {x^2}{2}+c_2 \\ \end{align*}