81.17.1 problem 21-8

Internal problem ID [21734]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 21. Applications of second order differential equations
Problem number : 21-8
Date solved : Thursday, October 02, 2025 at 08:01:32 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 b y^{\prime }+y&=k \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.062 (sec). Leaf size: 90

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

\[ y = \frac {-{\mathrm e}^{\left (-b +\sqrt {b^{2}-1}\right ) x} \left (b^{2}-1+\sqrt {b^{2}-1}\, b \right ) \left (-1+k \right )-\left (-\sqrt {b^{2}-1}\, b +b^{2}-1\right ) \left (-1+k \right ) {\mathrm e}^{-\left (b +\sqrt {b^{2}-1}\right ) x}+2 b^{2} k -2 k}{2 b^{2}-2} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 107

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

\begin{align*} y(x)&\to \frac {e^{-\left (\left (\sqrt {b^2-1}+b\right ) x\right )} \left (-b (k-1) \left (e^{2 \sqrt {b^2-1} x}-1\right )-\sqrt {b^2-1} \left ((k-1) e^{2 \sqrt {b^2-1} x}-2 k e^{\left (\sqrt {b^2-1}+b\right ) x}+k-1\right )\right )}{2 \sqrt {b^2-1}} \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 95

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(2*b*Derivative(y(x), x) - k + y(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)
 

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