67.8.42 problem 13.6 (h)

Internal problem ID [16537]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.6 (h)
Date solved : Thursday, October 02, 2025 at 01:36:11 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} 2 x y^{\prime } y^{\prime \prime }&={y^{\prime }}^{2}-1 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&=\sqrt {3} \\ \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 19

ode:=2*x*diff(y(x),x)*diff(diff(y(x),x),x) = diff(y(x),x)^2-1; 
ic:=[y(1) = 0, D(y)(1) = 3^(1/2)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (2 x +1\right )^{{3}/{2}}}{3}-\sqrt {3} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 26

ode=2*x*D[y[x],x]*D[y[x],{x,2}]==D[y[x],x]^2-1; 
ic={y[1]==0,Derivative[1][y][1]==Sqrt[3]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} \left ((2 x+1)^{3/2}-3 \sqrt {3}\right ) \end{align*}

✓ Sympy. Time used: 1.393 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2 + 1,0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): sqrt(3)} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (2 x + 1\right )^{\frac {3}{2}}}{3} - \sqrt {3} \]