23.4.15 problem 15

Internal problem ID [6317]
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 : 15
Date solved : Tuesday, September 30, 2025 at 02:46:43 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} a \,{\mathrm e}^{y}+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 28

ode:=a*exp(y(x))+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\ln \left (2\right )+\ln \left (\frac {\operatorname {sech}\left (\frac {x +c_2}{2 c_1}\right )^{2}}{c_1^{2} a}\right ) \]

✓ Mathematica. Time used: 22.23 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to \log \left (\frac {c_1 \text {sech}^2\left (\frac {1}{2} \sqrt {c_1 (x+c_2){}^2}\right )}{2 a}\right ) \end{align*}

✓ Sympy. Time used: 8.619 (sec). Leaf size: 78

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

\[ \left [ y{\left (x \right )} = \begin {cases} C_{1} C_{2} + C_{1} x & \text {for}\: a = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \log {\left (\frac {C_{1}}{a \cos {\left (C_{2} \sqrt {- C_{1}} + x \sqrt {- C_{1}} \right )} + a} \right )} & \text {for}\: a \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} C_{2} - C_{1} x & \text {for}\: a = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \log {\left (\frac {C_{1}}{a \cos {\left (C_{2} \sqrt {- C_{1}} - x \sqrt {- C_{1}} \right )} + a} \right )} & \text {for}\: a \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]