76.15.22 problem 23

Internal problem ID [17663]
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 : Tuesday, January 28, 2025 at 10:48:41 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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 65

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

\[ y = \frac {\left (-9 t^{3}-9 t^{2}-27 c_{1} -6 t -2\right ) {\mathrm e}^{-3 t}}{81}+\frac {2 t^{5}}{15}-\frac {2 t^{4}}{9}+\frac {8 t^{3}}{27}-\frac {8 t^{2}}{27}+\frac {16 t}{81}+c_{2} -\frac {\cos \left (3 t \right )}{18}-\frac {\sin \left (3 t \right )}{18} \]

✓ Solution by Mathematica

Time used: 1.123 (sec). Leaf size: 101

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

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