47.4.16 problem 17

Internal problem ID [9790]
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 : 17
Date solved : Tuesday, September 30, 2025 at 06:41:18 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 (0\right )&=1 \\ y^{\prime }\left (0\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 10

ode:=diff(diff(y(x),x),x) = x*diff(y(x),x)^2; 
ic:=[y(0) = 1, D(y)(0) = 1/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \operatorname {arctanh}\left (\frac {x}{2}\right )+1 \]

✓ Mathematica. Time used: 13.012 (sec). Leaf size: 43

ode=D[y[x],{x,2}]==x*(D[y[x],x])^2; 
ic={y[0]==1,Derivative[1][y][0] ==1/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x-\frac {2}{K[1]^2-4}dK[1]-\int _1^0-\frac {2}{K[1]^2-4}dK[1]+1 \end{align*}

✓ Sympy. Time used: 0.488 (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(0): 1, Subs(Derivative(y(x), x), x, 0): 1/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {\log {\left (x - 2 \right )}}{2} + \frac {\log {\left (x + 2 \right )}}{2} + 1 + \frac {i \pi }{2} \]