40.7.9 problem 18

Internal problem ID [6699]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 12. Linear equations of order n. Supplemetary problems. Page 81
Problem number : 18
Date solved : Sunday, March 30, 2025 at 11:20:05 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 39

ode:=y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2 = 2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {-2 c_1 x +2 x^{2}+2 c_2} \\ y &= -\sqrt {-2 c_1 x +2 x^{2}+2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 6.298 (sec). Leaf size: 101

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

\begin{align*} y(x)\to -\frac {\sqrt {4 (x+c_2){}^2-e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to \sqrt {2 (x+c_2){}^2-\frac {e^{2 c_1}}{2}} \\ y(x)\to -\sqrt {2} \sqrt {(x+c_2){}^2} \\ y(x)\to \sqrt {2} \sqrt {(x+c_2){}^2} \\ \end{align*}

✗ Sympy

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

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