Can someone help me please?Help Needed: Deriving 4th Order PDE in One Variable

[dglee]

For some reason i can't post in the calculus and beyond section
but i was hoping someone could help me with this question
eq.1 du/dt - fv = g*(dn/dx)
eq.2 dv/dt + fu = g*(dn/dg)
eq.3 du/dx+dv/dg=(1/(H)*(dn/dt)
manipulate the equations to derivate a single PDE in one variable n which is fourth order in time.
the hint was given eliminate u from (1) and (2) and from (1) and (3). Then eliminate v.
ok i was told to do this question by some sort of linear row reducing type of method
eq.1 (d/dt)*u - (f)*v = (g*d/dx)*n
eq.2 (f)*u + (d/dt)*v = (g*d/dg)*n
thus
eq.1 f*((d/dt)*u - (f)*v = (g*d/dx)*n)
eq.2 d/dt*((f)*u + (d/dt)*v = (g*d/dg)*n)
eq.1 (f*d/dt)*u - (f^2)*v = (f*g*d/dx)*n
eq.2 (f*d/dt)*u + (d^2/dt^2)*v = (g*d^2/dg*dt)*n
so i can eliminate u
then becomes
(-f^2-d/dt)*v = (f*g*d/dx - g*d^2/dg*dt)*n
so i did the same thing for 1 and 3. then i tried to eliminate v, but however i can't. I am not even sure if i even eliminiated u properly.

 

[AvroArrow]

Where did i go wrong?You haven't gone wrong - but it looks like you haven't used the hint given. The hint is to eliminate u from equations 1 and 2, and then eliminate v from equations 1 and 3. To eliminate u, start with equations 1 and 2 and subtract them from each other. This will give you an equation with just v on one side and terms related to n on the other. Then, to eliminate v, take that equation and subtract it from equation 1. This will leave you with an equation with just n on one side and terms related to derivatives of n on the other. That equation is the fourth-order PDE for n that you need.

 

[quicknote]

Sure, I can try to help you with this question. It looks like you are trying to derive a fourth-order partial differential equation (PDE) in one variable n using three given equations. First, let me clarify that the three equations you have given are not PDEs, but rather a system of three ordinary differential equations (ODEs) in the variables u, v, and n. To derive a fourth-order PDE in one variable, we need to eliminate all the other variables (u and v) from the equations.

Your approach of eliminating u and v separately is a good start. Let's take a closer look at the first equation:

eq.1 du/dt - fv = g*(dn/dx)

We can rewrite this equation as:

u' = fv + g*(dn/dx),

where u' denotes du/dt. Similarly, we can rewrite the second equation as:

v' = -fu + g*(dn/dg),

where v' denotes dv/dt. Now, to eliminate u, we can multiply the first equation by f and the second equation by -f and then add them together:

f*u' - f*v = f*g*(dn/dx) - f*g*(dn/dg)

This gives us:

f*u' - f*v = f*g*(dn/dx - dn/dg)

We can simplify this further by using the chain rule for derivatives:

f*u' - f*v = f*g*(d(n/dt))

where d(n/dt) denotes the total derivative of n with respect to t. Now, we can substitute this back into the original equations:

eq.1 f*u' - f*v = f*g*(d(n/dt))
eq.2 v' = -fu + g*(dn/dg)

This gives us a system of two equations in two variables (u' and v'). We can now eliminate v' by multiplying the first equation by f and the second equation by -f and then adding them together:

f^2*u' - f^2*v = f^2*g*(d(n/dt)) - f^2*g*(dn/dg)

This simplifies to:

f^2*u' - f^2*v = f^2*g*(d(n/dt - dn/dg))

We can rewrite this as:

f^2*u' = f^2*g*(d(n/dt - dn/dg)) + f^2