23.3.5 problem 5

Internal problem ID [5719]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 02:02:03 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=\operatorname {c1} \,{\mathrm e}^{a x}+\operatorname {c2} \,{\mathrm e}^{-b x} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x) = c1*exp(a*x)+c2/exp(b*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {c1} \,{\mathrm e}^{a x}}{a^{2}}+\frac {\operatorname {c2} \,{\mathrm e}^{-b x}}{b^{2}}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 33

ode=D[y[x],{x,2}] == c1*E^(a*x) + c2/E^(b*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\text {c1} e^{a x}}{a^2}+\frac {\text {c2} e^{-b x}}{b^2}+c_2 x+c_1 \end{align*}

✓ Sympy. Time used: 0.061 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c1 = symbols("c1") 
c2 = symbols("c2") 
y = Function("y") 
ode = Eq(-c1*exp(a*x) - c2*exp(-b*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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