76.15.27 problem 28

Internal problem ID [17668]
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 : 28
Date solved : Tuesday, January 28, 2025 at 10:50:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=t^{2} \sin \left (2 t \right )+\left (6 t +7\right ) \cos \left (2 t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 41

dsolve(diff(y(t),t$2)+4*y(t)=t^2*sin(2*t)+(6*t+7)*cos(2*t),y(t), singsol=all)
 

\[ y = \frac {\left (-8 t^{3}+96 c_{1} +39 t +84\right ) \cos \left (2 t \right )}{96}+\frac {13 \left (t^{2}+\frac {28}{13} t +\frac {16}{13} c_{2} -\frac {1}{4}\right ) \sin \left (2 t \right )}{16} \]

✓ Solution by Mathematica

Time used: 0.291 (sec). Leaf size: 51

DSolve[D[y[t],{t,2}]+4*y[t]==t^2*Sin[2*t]+(6*t+7)*Cos[2*t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \left (-\frac {t^3}{12}+\frac {13 t}{32}+\frac {7}{8}+c_1\right ) \cos (2 t)+\frac {1}{64} \left (104 t^2+224 t-13+128 c_2\right ) \sin (t) \cos (t) \]