88.15.6 problem 6

Internal problem ID [24102]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 4. Linear equations. Exercises at page 97
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:59:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=2 \,{\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)+diff(y(x),x)-2*y(x) = 2*exp(x); 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {7 \,{\mathrm e}^{x}}{9}+\frac {2 \,{\mathrm e}^{-2 x}}{9}+\frac {2 x \,{\mathrm e}^{x}}{3} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+D[y[x],x]-2*y[x]==2*Exp[x]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{9} e^{-2 x} \left (e^{3 x} (6 x+7)+2\right ) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = \left (\frac {2 x}{3} + \frac {7}{9}\right ) e^{x} + \frac {2 e^{- 2 x}}{9} \]