23.4.39 problem 39

Internal problem ID [6341]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 02:51:53 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 22

ode=D[y[x],{x,2}] == a^2 + b^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 b (x+c_1)))}{b^2} \end{align*}

✓ Sympy. Time used: 2.081 (sec). Leaf size: 65

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

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