Solving an IVP for a system of ODEs

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Discussion Overview

The discussion revolves around solving an initial value problem (IVP) for a system of ordinary differential equations (ODEs) involving two variables, u and v, with a particular focus on the role of a term w, which is not clearly defined as either a constant or a function of time.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Participants express confusion regarding the nature of w, questioning whether it is a constant or a function of time.
  • Some participants propose that w is a constant, while others suggest that it could be a function of t, leading to complications in solving the IVP.
  • One participant differentiates the second equation, leading to a new equation involving v'' and w, and questions the implications of treating w as a function of time.
  • Another participant notes that assuming w is a function complicates the problem due to the introduction of three functions and only two equations.
  • There is a suggestion that the problem might be solvable using Laplace transforms, though uncertainty remains about the appropriateness of this method.
  • A participant raises the possibility of w(t-5) being the Heaviside unit function, but acknowledges that this depends on the source of the problem.
  • The source of the problem is identified as a textbook, with specific details provided about the title and page number.

Areas of Agreement / Disagreement

Participants do not reach consensus on the nature of w, with multiple competing views regarding whether it is a constant or a function of time. The discussion remains unresolved regarding the best approach to solving the IVP.

Contextual Notes

Participants note that the problem may become more complex if w is treated as a function of time, leading to additional variables without sufficient equations to solve them. There is also mention of potential limitations in the mathematical steps taken, particularly regarding the use of Laplace transforms.

krish
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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.
 
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What is $$w$$ ? ,is it a constant or a function of $t$ ?
 
ZaidAlyafey said:
What is $$w$$ ? ,is it a constant or a function of $t$ ?

I believe w is a constant.
 
krish said:
I believe w is a constant.

Aha , try differentiating one of the equations and tell me if you got any ideas .
 
ZaidAlyafey said:
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.
 
krish said:
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 ?
 
ZaidAlyafey said:
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?
 
krish said:
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 ?
 
Last edited:
ZaidAlyafey said:
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.
 
  • #10
krish said:
It's in the review questions in my Differential Equations textbook.

Ok , tell me the name of the textbook and the page number .
 
  • #11
ZaidAlyafey said:
Ok , tell me the name of the textbook and the page number .

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

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