15.18.40 problem 40

Internal problem ID [3283]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 40
Date solved : Sunday, March 30, 2025 at 01:31:16 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

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

✓ Maple. Time used: 0.082 (sec). Leaf size: 16

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

\[ y = \frac {1}{-\sqrt {3}\, \sin \left (x \right )+\cos \left (x \right )} \]

✓ Mathematica. Time used: 0.201 (sec). Leaf size: 19

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

\[ y(x)\to \frac {1}{2} \csc \left (\frac {1}{6} (\pi -6 x)\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2)*sqrt((-y(x) + Derivative(y(x), (x, 2)))*y(x))/2 + Derivative(y(x), x) cannot be solved by the factorable group method