Solving Non-Linear Differential Equations for Free Fall Due to Gravity

[Pete69]

Homework Statement

I was given a problem to solve for the speed of a body falling under gravity [equation (1)] where g is acceleration due to gravity, which was easy enough.. but then i thought i would extend it to the case where g is non-constant, and so arrived at equation (2), (where where z is the height above Earth [z'=dz/dt and z=dv/dt and z^-2 means z to power -2], and M is the mass of the Earth and G is the gravitational constant)

Homework Equations

(1) : dv/dt = - g - kv

(2) : z'' + kz' + GMz^-2 = 0

The Attempt at a Solution

I believe this is a non-linear second order DE?? i attempted to solve by setting

z'' + kz' = 0

and solving the complimentary equation, which was OK, but when i came to solve for the particular integral

z'' + kz' = -GMz^-2

i ran into problems, as after substituting in the D and Q operators (http://silmaril.math.sci.qut.edu.au/~gustafso/mab112/topic12/ ), i could not use the First Shift Theorem, as the RHS is not in an exponential form...

Any ideas anyone?

 

[quantumdude]

I believe this is a non-linear second order DE?? i attempted to solve by setting

Yes, but rather than get into that let's look at your first order equation for v.

Do the following.

1.) Show that \frac{dv}{dt}=v\frac{dv}{dz}.
2.) Insert g=\frac{GM}{z^2} into the equation.
3.) Find an integrating factor that makes this equation exact (it can be done).
4.) Solve.

 

[HallsofIvy]

Homework Statement

I was given a problem to solve for the speed of a body falling under gravity [equation (1)] where g is acceleration due to gravity, which was easy enough.. but then i thought i would extend it to the case where g is non-constant, and so arrived at equation (2), (where where z is the height above Earth [z'=dz/dt and z=dv/dt and z^-2 means z to power -2], and M is the mass of the Earth and G is the gravitational constant)

Homework Equations

(1) : dv/dt = - g - kv

(2) : z'' + kz' + GMz^-2 = 0

The Attempt at a Solution

I believe this is a non-linear second order DE?? i attempted to solve by setting

z'' + kz' = 0

and solving the complimentary equation, which was OK, but when i came to solve for the particular integral

There is no "complementary equation" nor is there a "particular integral". Those are both concepts in linear differential equations where the non-homogeneous part is a function of the independent variable only.

z'' + kz' = -GMz^-2

i ran into problems, as after substituting in the D and Q operators (http://silmaril.math.sci.qut.edu.au/~gustafso/mab112/topic12/ ), i could not use the First Shift Theorem, as the RHS is not in an exponential form...

Any ideas anyone?