79.13.2 problem (b)

Internal problem ID [21115]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter IV. Linear Differential Equations. Excercise VI at page 209
Problem number : (b)
Date solved : Thursday, October 02, 2025 at 07:08:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&={\mathrm e}^{x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+5*y(x) = exp(x); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{x} \left (2 \sin \left (2 x \right )-3 \cos \left (2 x \right )-1\right )}{4} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 26

ode=D[y[x],{x,2}]-2*D[y[x],x]+5*y[x]==Exp[x]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} e^x (-2 \sin (2 x)+3 \cos (2 x)+1) \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) - exp(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- \frac {\sin {\left (2 x \right )}}{2} + \frac {3 \cos {\left (2 x \right )}}{4} + \frac {1}{4}\right ) e^{x} \]