# Math Help - could someone give me advice? differential equation.

1. ## could someone give me advice? differential equation.

Hi im having problems with the following 2 question and was wondering if somebody could give me some help on how to do it. Im not the best at math so any help/advice/answer would be greatly appreciated. Feel free to move this thread if its in the wrong catagory.
Totster.
1. (a) Solve the following second order linear differential equation

(d^2(
x)/(d)t^2)+ 5(dx/dt) + 4x = 0

subject to the initial condition
x(0) = 0 and x˙(0) = 1. What type
of damping does this motion illustrate?
(b) Solve the following second order linear differential equation

(d^2(
x)/(d)t^2)+ x = cos(t)
subject to the initial condition
x(0) = 0 and x˙ (0) = 0. What type
of motion do you find?
2. A satellite undergoes an elliptic orbit about the Earth of mass
M, with
maximum distance 6
R and minimum distance 3R from the Earth’s
centre where
R is the radius of the Earth.
(a) Show that the minimum velocity
vmin and the maximum velocity

v
max are given by

v
max = 2vmin = 2/3GM/R

(b) Show that the eccentricity of the satellite is
e = 1/3.

Thanks

2. Originally Posted by totster
Hi im having problems with the following 2 question and was wondering if somebody could give me some help on how to do it. Im not the best at math so any help/advice/answer would be greatly appreciated. Feel free to move this thread if its in the wrong catagory.
Totster.
1. (a) Solve the following second order linear differential equation

(d^2(
x)/(d)t^2)+ 5(dx/dt) + 4x = 0

subject to the initial condition
x(0) = 0 and x˙(0) = 1. What type
of damping does this motion illustrate?

The characteristic equation is $r^2+ 5r+ 4= 0$. Can you solve that? What do the solutions of that equation tell you about the solution to the differential equaiton?

(b) Solve the following second order linear differential equation
(d^2(
x)/(d)t^2)+ x = cos(t)
subject to the initial condition
x(0) = 0 and x˙ (0) = 0. What type
of motion do you find?
2. A satellite undergoes an elliptic orbit about the Earth of mass
M, with
maximum distance 6
R and minimum distance 3R from the Earth’s
centre where
R is the radius of the Earth.
(a) Show that the minimum velocity
vmin and the maximum velocity

v
max are given by

v
max = 2vmin = 2/3GM/R

(b) Show that the eccentricity of the satellite is
e = 1/3.

Thanks