The differential equation describing the truck movement is a linear non homogeneous differential equation
(dv)/(dt) + lambda v =e^(-alpha/3t)dvdt+λv=e−α3t
The solution for those type of equations can be obtained as the sum of two solutions. The solution to the homogeneous equation
(dv_h)/(dt)+lambda v_h = 0dvhdt+λvh=0
plus the particular solution
(dv_p)/(dt) + lambda v_p =e^(-alpha/3t)dvpdt+λvp=e−α3t
After that, v = v_h + v_pv=vh+vp
Obtaining v_hvh is quite easy
We propose v_h = C e^(xi t)vh=Ceξt then substituting into the homogeneous
lambda C e^(xi t)+xi C e^(xi t)=C(lambda+ xi)e^(xi t)=0λCeξt+ξCeξt=C(λ+ξ)eξt=0
This condition is satisfied for lambda + xi = 0λ+ξ=0 or
xi = -lambdaξ=−λ
and v_h = C e^(-lambda t)vh=Ce−λt
The particular is obtained supposing that C = C(t)C=C(t) and introducing into the complete equation
d/(dt)(C(t)e^(-lambda t))+lambda C(t)e^(-lambda t)=e^(-alpha/3t)ddt(C(t)e−λt)+λC(t)e−λt=e−α3t
so we obtain
(dC)/(dt)e^(-lambda t)=e^(-alpha/3 t)dCdte−λt=e−α3t
then
(dC)/(dt) = e^((lambda-alpha/3)t)dCdt=e(λ−α3)t and integrating
C(t) = e^((lambda-alpha/3)t)/(lambda-alpha/3)C(t)=e(λ−α3)tλ−α3
and finally
v = C_0 e^(-lambda t)+e^((lambda-alpha/3)t)/(lambda-alpha/3)e^(-lambda t) = C_0 e^(-lambda t)+e^(-alpha/3t)/(lambda-alpha/3)v=C0e−λt+e(λ−α3)tλ−α3e−λt=C0e−λt+e−α3tλ−α3
