Internal problem ID [6082]
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: 16.
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 {y^{\prime \prime }-x \left (y^{\prime }\right )^{2}=0} \end {gather*} With initial conditions \begin {align*} \left [y \relax (2) = \frac {\pi }{4}, y^{\prime }\relax (2) = -{\frac {1}{4}}\right ] \end {align*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 8
dsolve([diff(y(x),x$2)=x*diff(y(x),x)^2,y(2) = 1/4*Pi, D(y)(2) = -1/4],y(x), singsol=all)
\[ y \relax (x ) = \mathrm {arccot}\left (\frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 1.616 (sec). Leaf size: 19
DSolve[{y''[x]==x*(y'[x])^2,{y[2]==1/4*Pi,y'[2]==-1/4}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\pi -2 \text {ArcTan}\left (\frac {x}{2}\right )\right ) \\ \end{align*}