83.31.5 problem 5

Internal problem ID [19318]
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 : 5
Date solved : Thursday, March 13, 2025 at 02:13:43 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 18

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

\[ y \left (x \right ) = -\frac {\ln \left (-a \left (c_{1} x +c_{2} \right )\right )}{a} \]

✓ Mathematica. Time used: 0.208 (sec). Leaf size: 20

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

\[ y(x)\to c_2-\frac {\log (a x+c_1)}{a} \]

✓ Sympy. Time used: 0.542 (sec). Leaf size: 12

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

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