Internal problem ID [5488]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.9. Reduction of Order. Page
38
Problem number: 1(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {2 y y^{\prime \prime }-1-\left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.098 (sec). Leaf size: 22
dsolve(2*y(x)*diff(y(x),x$2)=1+(diff(y(x),x))^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (c_{1}^{2}+1\right ) x^{2}}{4 c_{2}}+c_{1} x +c_{2} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 29
DSolve[2*y[x]*y''[x]==1+(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (1+c_1{}^2\right ) x^2}{4 c_2}+c_1 x+c_2 \\ \end{align*}