76.15.22 problem 23

Internal problem ID [17584]
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, March 13, 2025 at 10:14:47 AM
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: 0.002 (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: 1.123 (sec). Leaf size: 101

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]
 

\[ y(t)\to \frac {1}{810} \left (2 e^{-3 t} \left (54 e^{3 t} t^5-90 e^{3 t} t^4+15 \left (8 e^{3 t}-3\right ) t^3-15 \left (8 e^{3 t}+3\right ) t^2+10 \left (8 e^{3 t}-3\right ) t-5 (2+27 c_1)\right )-45 \sin (3 t)-45 \cos (3 t)\right )+c_2 \]

✓ Sympy. Time used: 0.536 (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)
 

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