60.2.36 problem Problem 51

Internal problem ID [15216]
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 51
Date solved : Thursday, October 02, 2025 at 10:07:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+10 x^{\prime }+25 x&=2^{t}+t \,{\mathrm e}^{-5 t} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 40

ode:=diff(diff(x(t),t),t)+10*diff(x(t),t)+25*x(t) = 2^t+t*exp(-5*t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {\left (\ln \left (2\right )+5\right )^{2} \left (t^{3}+6 t c_1 +6 c_2 \right ) {\mathrm e}^{-5 t}+6 \,2^{t}}{6 \left (\ln \left (2\right )+5\right )^{2}} \]

✓ Mathematica. Time used: 0.179 (sec). Leaf size: 72

ode=D[x[t],{t,2}]+10*D[x[t],t]+25*x[t]==2^t+t*Exp[-5*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {e^{-5 t} \left (t^3 \left (25+\log ^2(2)+\log (1024)\right )+3\ 2^{t+1} e^{5 t}+c_2 t \left (150+6 \log ^2(2)+\log (1152921504606846976)\right )+c_1 \left (150+6 \log ^2(2)+\log (1152921504606846976)\right )\right )}{6 (5+\log (2))^2} \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-2**t - t*exp(-5*t) + 25*x(t) + 10*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {2^{t}}{\log {\left (2 \right )}^{2} + 10 \log {\left (2 \right )} + 25} + \left (C_{1} + t \left (C_{2} + \frac {t^{2}}{6}\right )\right ) e^{- 5 t} \]