83.32.6 problem 6

Internal problem ID [19326]
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 (F) at page 113
Problem number : 6
Date solved : Thursday, March 13, 2025 at 02:16:01 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.242 (sec). Leaf size: 22

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

\[ y(x)\to c_2-\frac {\log (\cos (a k (x+c_1)))}{k^2} \]

✓ Sympy. Time used: 3.465 (sec). Leaf size: 65

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

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