90.15.9 problem 9

Internal problem ID [25256]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 3. Second Order Constant Coefficient Linear Differential Equations. Exercises at page 251
Problem number : 9
Date solved : Thursday, October 02, 2025 at 11:59:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-8 y^{\prime }+25 y&=36 t \,{\mathrm e}^{4 t} \sin \left (3 t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(y(t),t),t)-8*diff(y(t),t)+25*y(t) = 36*t*exp(4*t)*sin(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -3 \,{\mathrm e}^{4 t} \left (\left (t^{2}-\frac {c_1}{3}\right ) \cos \left (3 t \right )-\frac {\sin \left (3 t \right ) \left (t +c_2 \right )}{3}\right ) \]

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 41

ode=D[y[t],{t,2}]-8*D[y[t],{t,1}]+25*y[t]==36*t*Exp[4*t]*Sin[3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{6} e^{4 t} \left (\left (-18 t^2+1+6 c_2\right ) \cos (3 t)+6 (t+c_1) \sin (3 t)\right ) \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 87

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-36*t*exp(4*t)*sin(3*t) + 25*y(t) - 7*Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{- \frac {5 \sqrt {7} t}{7}} + C_{2} e^{\frac {5 \sqrt {7} t}{7}} - \frac {3 t e^{4 t} \sin {\left (3 t \right )}}{100} + \frac {21 t e^{4 t} \cos {\left (3 t \right )}}{100} - \frac {21 e^{4 t} \sin {\left (3 t \right )}}{400} - \frac {7 e^{4 t} \cos {\left (3 t \right )}}{100} \]