85.69.6 problem 6

Internal problem ID [22922]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. C Exercises at page 217
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:16:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} Q^{\prime \prime }+k Q&=e \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.066 (sec). Leaf size: 61

ode:=diff(diff(Q(t),t),t)+k*Q(t) = e(t); 
ic:=[Q(0) = q__0, D(Q)(0) = 0]; 
dsolve([ode,op(ic)],Q(t), singsol=all);
 

\[ Q = \frac {\int _{0}^{t}\cos \left (\sqrt {k}\, \textit {\_z1} \right ) e \left (\textit {\_z1} \right )d \textit {\_z1} \sin \left (\sqrt {k}\, t \right )-\cos \left (\sqrt {k}\, t \right ) \left (-q_{0} \sqrt {k}+\int _{0}^{t}\sin \left (\sqrt {k}\, \textit {\_z1} \right ) e \left (\textit {\_z1} \right )d \textit {\_z1} \right )}{\sqrt {k}} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 140

ode=D[Q[t],{t,2}]+k*Q[t]==e[t]; 
ic={Q[0]==q0,Derivative[1][Q][0] ==0}; 
DSolve[{ode,ic},Q[t],t,IncludeSingularSolutions->True]
 

\begin{align*} Q(t)&\to \cos \left (\sqrt {k} t\right ) \left (\int _1^t-\frac {e(K[1]) \sin \left (\sqrt {k} K[1]\right )}{\sqrt {k}}dK[1]-\int _1^0-\frac {e(K[1]) \sin \left (\sqrt {k} K[1]\right )}{\sqrt {k}}dK[1]+\text {q0}\right )-\sin \left (\sqrt {k} t\right ) \int _1^0\frac {\cos \left (\sqrt {k} K[2]\right ) e(K[2])}{\sqrt {k}}dK[2]+\sin \left (\sqrt {k} t\right ) \int _1^t\frac {\cos \left (\sqrt {k} K[2]\right ) e(K[2])}{\sqrt {k}}dK[2] \end{align*}

✓ Sympy. Time used: 0.527 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
k = symbols("k") 
q = Function("q") 
e = Function("e") 
ode = Eq(k*q(t) - e(t) + Derivative(q(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=q(t),ics=ics)
 

\[ q{\left (t \right )} = \left (C_{1} + \frac {\int e{\left (t \right )} e^{- t \sqrt {- k}}\, dt}{2 \sqrt {- k}}\right ) e^{t \sqrt {- k}} + \left (C_{2} - \frac {\int e{\left (t \right )} e^{t \sqrt {- k}}\, dt}{2 \sqrt {- k}}\right ) e^{- t \sqrt {- k}} \]