76.17.23 problem 34 (b)

Internal problem ID [17638]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 34 (b)
Date solved : Monday, March 31, 2025 at 04:23:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=g \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 42

ode:=diff(diff(y(t),t),t)+y(t) = g(t); 
ic:=y(0) = y__0, D(y)(0) = y__1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \sin \left (t \right ) y_{1} +\cos \left (t \right ) y_{0} +\int _{0}^{t}\cos \left (\textit {\_z1} \right ) g \left (\textit {\_z1} \right )d \textit {\_z1} \sin \left (t \right )-\int _{0}^{t}\sin \left (\textit {\_z1} \right ) g \left (\textit {\_z1} \right )d \textit {\_z1} \cos \left (t \right ) \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 86

ode=D[y[t],{t,2}]+y[t]==g[t]; 
ic={y[0]==y0,Derivative[1][y][0] == y1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\cos (t) \int _1^0-g(K[1]) \sin (K[1])dK[1]+\cos (t) \int _1^t-g(K[1]) \sin (K[1])dK[1]-\sin (t) \int _1^0\cos (K[2]) g(K[2])dK[2]+\sin (t) \int _1^t\cos (K[2]) g(K[2])dK[2]+\text {y0} \cos (t)+\text {y1} \sin (t) \]

✓ Sympy. Time used: 0.628 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-g(t) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): y__0, Subs(Derivative(y(t), t), t, 0): y__1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (y^{0} - \int g{\left (t \right )} \sin {\left (t \right )}\, dt + \int \limits ^{0} g{\left (t \right )} \sin {\left (t \right )}\, dt\right ) \cos {\left (t \right )} + \left (y^{1} + \int g{\left (t \right )} \cos {\left (t \right )}\, dt - \int \limits ^{0} g{\left (t \right )} \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )} \]