70.15.22 problem 23

Internal problem ID [18950]
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 : 23
Date solved : Thursday, October 02, 2025 at 03:33:18 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 65

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t) = 2*t^4+t^2*exp(-3*t)+sin(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (-9 t^{3}-9 t^{2}-27 c_1 -6 t -2\right ) {\mathrm e}^{-3 t}}{81}+\frac {2 t^{5}}{15}-\frac {2 t^{4}}{9}+\frac {8 t^{3}}{27}-\frac {8 t^{2}}{27}+\frac {16 t}{81}+c_2 -\frac {\cos \left (3 t \right )}{18}-\frac {\sin \left (3 t \right )}{18} \]

✓ Mathematica. Time used: 3.73 (sec). Leaf size: 62

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

\begin{align*} y(t)&\to \int _1^te^{-3 K[2]} \left (c_1+\int _1^{K[2]}\left (2 e^{3 K[1]} K[1]^4+K[1]^2+e^{3 K[1]} \sin (3 K[1])\right )dK[1]\right )dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 0.331 (sec). Leaf size: 71

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

\[ y{\left (t \right )} = C_{1} + \frac {2 t^{5}}{15} - \frac {2 t^{4}}{9} + \frac {8 t^{3}}{27} - \frac {8 t^{2}}{27} + \frac {16 t}{81} + \left (C_{2} - \frac {t^{3}}{9} - \frac {t^{2}}{9} - \frac {2 t}{27}\right ) e^{- 3 t} - \frac {\sin {\left (3 t \right )}}{18} - \frac {\cos {\left (3 t \right )}}{18} \]