53.4.15 problem 16

Internal problem ID [8503]
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
Date solved : Monday, January 27, 2025 at 04:07:35 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }&=x {y^{\prime }}^{2} \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=\frac {\pi }{4}\\ y^{\prime }\left (2\right )&=-{\frac {1}{4}} \end{align*}

✓ Solution by Maple

Time used: 0.055 (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)
 

\[ y = \operatorname {arccot}\left (\frac {x}{2}\right ) \]

✓ Solution by Mathematica

Time used: 1.273 (sec). Leaf size: 19

DSolve[{D[y[x],{x,2}]==x*(D[y[x],x])^2,{y[2]==1/4*Pi,Derivative[1][y][2]==-1/4}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (\pi -2 \arctan \left (\frac {x}{2}\right )\right ) \]