76.12.10 problem 16

Internal problem ID [17487]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 16
Date solved : Thursday, March 13, 2025 at 10:10:15 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 33

ode:=y(t)*diff(diff(y(t),t),t)+diff(y(t),t)^2 = 0; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \sqrt {2 c_{1} t +2 c_{2}} \\ y &= -\sqrt {2 c_{1} t +2 c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.16 (sec). Leaf size: 20

ode=y[t]*D[y[t],{t,2}]+D[y[t],t]^2==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to c_2 \sqrt {2 t-c_1} \]

✗ Sympy

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

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