76.48.21 problem Ex. 21

Internal problem ID [20287]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. End of chapter problems at page 107
Problem number : Ex. 21
Date solved : Thursday, October 02, 2025 at 05:41:02 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: 0.021 (sec). Leaf size: 33

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

\[ y = -\frac {\ln \left (\frac {k \left (c_1 \tan \left (k a x \right )-c_2 \right )}{a \sec \left (k a x \right )}\right )}{k^{2}} \]

✓ Mathematica. Time used: 0.189 (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]
 

\begin{align*} y(x)&\to c_2-\frac {\log (\cos (a k (x+c_1)))}{k^2} \end{align*}

✓ Sympy. Time used: 2.141 (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)
 

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