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 : Sunday, March 30, 2025 at 01:12:18 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*}

✓ Maple. Time used: 0.087 (sec). Leaf size: 8

ode:=diff(diff(y(x),x),x) = x*diff(y(x),x)^2; 
ic:=y(2) = 1/4*Pi, D(y)(2) = -1/4; 
dsolve([ode,ic],y(x), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 0.818 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {y(2): pi/4, Subs(Derivative(y(x), x), x, 2): -1/4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {i \log {\left (x - 2 i \right )}}{2} - \frac {i \log {\left (x + 2 i \right )}}{2} \]