Question

Could someone help me with this problem , please

Answer 1

2nd order non-hom D.E.: you will need particular solution and complementary solution

so for complementary solution, first solve:
\(m\ddot x + b \ \dot x = 0\) or \(\ddot x + \frac{b}{m} \ \dot x = 0\)
using diff operators:
\(D(D+\frac{b}{m}) x = 0 \)
so you should get a constant, \(c_1\), and constant \(c_2\) times an exponential \(e^{-bt/m}\)

for the particular solution, start with \(x(t) = A t^2 + Bt + C\) and you get \(\dot x = 2 A t + B\) and \(\ddot x = 2A\)

then stuff these back into \(\ddot x + \frac{b}{m} \ \dot x = g\) to find A, B and C

then add them up and apply the boundary conditions, \(x(0) = 0\) and \(\dot x(+\infty) = 0\) to find the \(c_1\) and \(c_2\).

those are the steps.

Answer 2

Ah very nice @IrishBoy123