66.2.8 problem Problem 8

Internal problem ID [13907]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 8
Date solved : Tuesday, January 28, 2025 at 06:08:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-4 x^{\prime }+4 x&={\mathrm e}^{t}+{\mathrm e}^{2 t}+1 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 27

dsolve(diff(x(t),t$2)-4*diff(x(t),t)+4*x(t)=exp(t)+exp(2*t)+1,x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (4 c_{1} t +2 t^{2}+4 c_{2} \right ) {\mathrm e}^{2 t}}{4}+{\mathrm e}^{t}+\frac {1}{4} \]

✓ Solution by Mathematica

Time used: 0.281 (sec). Leaf size: 32

DSolve[D[x[t],{t,2}]-4*D[x[t],t]+4*x[t]==Exp[t]+Exp[2*t]+1,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{2 t} \left (\frac {t^2}{2}+c_2 t+c_1\right )+e^t+\frac {1}{4} \]