83.31.4 problem 4

Internal problem ID [19325]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (E) at page 112
Problem number : 4
Date solved : Monday, March 31, 2025 at 07:07:00 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+4 {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+4*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\ln \left (2\right )}{4}+\frac {\ln \left (-c_1 \,{\mathrm e}^{-2 x}+2 c_2 \right )}{4} \]

✓ Mathematica. Time used: 0.384 (sec). Leaf size: 65

ode=D[y[x],{x,2}]+2*D[y[x],x]+4*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{4} \left (-\log \left (e^{2 x}\right )+\log \left (-e^{2 x}+2 e^{c_1}\right )\right )+c_2 \\ y(x)\to \frac {1}{4} \left (\log \left (-e^{2 x}\right )-\log \left (e^{2 x}\right )+4 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 1.202 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*Derivative(y(x), x)**2 + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} - \frac {x}{2} + \frac {\log {\left (C_{2} + e^{2 x} \right )}}{4} \]