74.13.39 problem 67

Internal problem ID [16334]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number : 67
Date solved : Monday, March 31, 2025 at 02:50:31 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 26

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

\begin{align*} y &= 0 \\ y &= \sqrt {c_{1}^{2}+c_{2}^{2}}+c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.607 (sec). Leaf size: 44

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

\[ y(t)\to c_2 \exp \left (\int _1^t\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [c_1-\frac {K[2]}{2}\right ]dK[2]\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t)**2 + 2*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) + 2*Derivative(y(t), (t, 2)))*y(t)) + Derivative(y(t), t) cannot be solved by the factorable group method