89.24.19 problem 19

Internal problem ID [24856]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Exercises at page 171
Problem number : 19
Date solved : Thursday, October 02, 2025 at 10:48:44 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y-3 y^{\prime }+y^{\prime \prime }&=15 \sqrt {1+{\mathrm e}^{-x}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)-3*diff(y(x),x)+2*y(x) = 15*(1+exp(-x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (4 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{x}+4\right ) \sqrt {1+{\mathrm e}^{-x}}+{\mathrm e}^{2 x} c_1 +{\mathrm e}^{x} c_2 \]

✓ Mathematica. Time used: 0.283 (sec). Leaf size: 40

ode=D[y[x],{x,2}]-3*D[y[x],x]+2*y[x]==  15*Sqrt[1+Exp[-x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 4 \sqrt {e^{-x}+1} \left (e^x+1\right )^2+c_1 e^x+c_2 e^{2 x} \end{align*}

✓ Sympy. Time used: 12.829 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-15*sqrt(1 + exp(-x)) + 2*y(x) - 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + 10 \left (1 + e^{- x}\right )^{\frac {3}{2}} + \left (C_{2} + 15 \int \sqrt {1 + e^{- x}} e^{- 2 x}\, dx\right ) e^{x}\right ) e^{x} \]