70.15.11 problem 11

Internal problem ID [18939]
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, October 02, 2025 at 03:33:10 PM
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: 0.003 (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);
 

\[ y = -2 \,{\mathrm e}^{-t} c_1 +14-6 t +t^{2}-\frac {9 \cos \left (t \right )}{10}-\frac {3 \sin \left (t \right )}{10}+{\mathrm e}^{-\frac {t}{2}} c_2 \]

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 81

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]
 

\begin{align*} y(t)&\to e^{-t} \left (e^{t/2} \int _1^te^{\frac {K[1]}{2}} \left (K[1]^2+3 \sin (K[1])\right )dK[1]+\int _1^t-e^{K[2]} \left (K[2]^2+3 \sin (K[2])\right )dK[2]+c_1 e^{t/2}+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.145 (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)
 

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