76.17.4 problem 13

Internal problem ID [17619]
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 : 13
Date solved : Monday, March 31, 2025 at 04:22:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }-4 y^{\prime }+y&=16 \,{\mathrm e}^{\frac {t}{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 19

ode:=4*diff(diff(y(t),t),t)-4*diff(y(t),t)+y(t) = 16*exp(1/2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{\frac {t}{2}} \left (t c_1 +2 t^{2}+c_2 \right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 25

ode=4*D[y[t],{t,2}]-4*D[y[t],t]+y[t]==16*Exp[t/2]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{t/2} \left (2 t^2+c_2 t+c_1\right ) \]

✓ Sympy. Time used: 0.194 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 16*exp(t/2) - 4*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} + t \left (C_{2} + 2 t\right )\right ) e^{\frac {t}{2}} \]