76.15.11 problem 11

Internal problem ID [17573]
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 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 11
Date solved : Thursday, March 13, 2025 at 10:13:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y^{\prime \prime }+3 y^{\prime }+y&=t^{2}+3 \sin \left (t \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 33

ode:=2*diff(diff(y(t),t),t)+3*diff(y(t),t)+y(t) = t^2+3*sin(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -2 \,{\mathrm e}^{-t} c_{1} +14-6 t +t^{2}-\frac {9 \cos \left (t \right )}{10}-\frac {3 \sin \left (t \right )}{10}+c_{2} {\mathrm e}^{-\frac {t}{2}} \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 43

ode=2*D[y[t],{t,2}]+3*D[y[t],t]+y[t]==t^2+3*Sin[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to t^2-6 t-\frac {3 \sin (t)}{10}-\frac {9 \cos (t)}{10}+c_1 e^{-t/2}+c_2 e^{-t}+14 \]

✓ Sympy. Time used: 0.237 (sec). Leaf size: 36

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

\[ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{- \frac {t}{2}} + t^{2} - 6 t - \frac {3 \sin {\left (t \right )}}{10} - \frac {9 \cos {\left (t \right )}}{10} + 14 \]