## anonymous one year ago Could someone help me with this problem , please

1. anonymous

2. IrishBoy123

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.

3. Astrophysics

Ah very nice @IrishBoy123