58.20.10 problem 10

Internal problem ID [14934]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 10
Date solved : Thursday, October 02, 2025 at 09:56:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-8 y^{\prime }+15 y&=9 \,{\mathrm e}^{2 t} t \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5 \\ y^{\prime }\left (0\right )&=10 \\ \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)-8*diff(y(t),t)+15*y(t) = 9*t*exp(2*t); 
ic:=[y(0) = 5, D(y)(0) = 10]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 28

ode=D[y[t],{t,2}]-8*D[y[t],t]+15*y[t]==9*t*Exp[2*t]; 
ic={y[0]==5,Derivative[1][y][0]==10}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{2 t} \left (3 t+3 e^t-2 e^{3 t}+4\right ) \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*t*exp(2*t) + 15*y(t) - 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 5, Subs(Derivative(y(t), t), t, 0): 10} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (3 t - 2 e^{3 t} + 3 e^{t} + 4\right ) e^{2 t} \]