14.11.3 problem 3

Internal problem ID [2596]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.5. Method of judicious guessing. Excercises page 164
Problem number : 3
Date solved : Monday, January 27, 2025 at 06:01:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=t^{2} {\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 35

DSolve[D[y[t],{t,2}]-y[t]==t*Exp[t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{8} e^t \left (2 t^2-2 t+1+8 c_1\right )+c_2 e^{-t} \]