76.17.11 problem 20

Internal problem ID [17626]
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 : 20
Date solved : Monday, March 31, 2025 at 04:22:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+6 y&=g \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)+6*y(t) = g(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{2 t} \left (\left (\int g \left (t \right ) {\mathrm e}^{-3 t}d t +c_2 \right ) {\mathrm e}^{t}-\int g \left (t \right ) {\mathrm e}^{-2 t}d t +c_1 \right ) \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 59

ode=D[y[t],{t,2}]-5*D[y[t],t]+6*y[t]==g[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{2 t} \left (\int _1^t-e^{-2 K[1]} g(K[1])dK[1]+e^t \int _1^te^{-3 K[2]} g(K[2])dK[2]+c_2 e^t+c_1\right ) \]

✓ Sympy. Time used: 0.859 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-g(t) + 6*y(t) - 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} + \left (C_{2} + \int g{\left (t \right )} e^{- 3 t}\, dt\right ) e^{t} - \int g{\left (t \right )} e^{- 2 t}\, dt\right ) e^{2 t} \]