Solving an IVP for a system of ODEs

[krish]

Hello, I am having trouble solving the below IVP, particularly I am confused with the w:

du/dt = v - w(t-5)

dv/dt = 2 - u(t)

u(0)=0, v(0)=0

Any help would be great. Thank you.

 

[alyafey22]

What is $$w$$ ? ,is it a constant or a function of $t$ ?

 

[krish]

What is $$w$$ ? ,is it a constant or a function of $t$ ?

I believe w is a constant.

 

[alyafey22]

I believe w is a constant.

Aha , try differentiating one of the equations and tell me if you got any ideas .

 

[krish]

Aha , try differentiating one of the equations and tell me if you got any ideas .

So I differentiated the second equation with respect to t:
v'' = -du/dt

Then I substitute first equation for du/dt:

v'' = -(v - w(t-5)) = -v + w(t-5)
v'' + v - w(t-5) = 0

Does it become: v'' + v = wt - 5w ? How does keeping the w matter?

And then I solve the IVP. Is this correct? But what if w is a function of t? Then I am confused, is that possible? Thank you for answering.

 

[alyafey22]

So I differentiated the second equation with respect to t:
v'' = -du/dt

Then I substitute first equation for du/dt:

v'' = -(v - w(t-5)) = -v + w(t-5)
v'' + v - w(t-5) = 0

Does it become: v'' + v = wt - 5w ? How does keeping the w matter?

And then I solve the IVP. Is this correct? But what if w is a function of t? Then I am confused, is that possible? Thank you for answering.

That seems a not-easy problem to deal with if we try the other differentiation we get

$$u''+ut = 2- w $$

The problem will get more complicated if assumed that $w$ a function because we have a three functions and two equations !

- - - Updated - - -

It might be solvable by Laplace but I don't know whether it is an acceptable solution ?

 

[krish]

That seems a not-easy problem to deal with if we try the other differentiation we get

$$u''+ut = 2- w $$

The problem will get more complicated if assumed that $w$ a function because we have a three functions and two equations !

- - - Updated - - -

It might be solvable by Laplace but I don't know whether it is an acceptable solution ?

Can w(t-5) be the Heavyside unit function?

 

[alyafey22]

Can w(t-5) be the Heavyside unit function?

I don't know there is no indication , that depends on the source.
From Where did you get that problem ?

 

[krish]

I don't know there is no indication , that depends on the source.
From Where did you get that problem ?

It's in the review questions in my Differential Equations textbook.

 

[alyafey22]

It's in the review questions in my Differential Equations textbook.

Ok , tell me the name of the textbook and the page number .

 

[krish]

Ok , tell me the name of the textbook and the page number .

Differential Equations (From Engineering Viewpoint) - Dr. R.C. Shah
Page 309