[SOLVED]Solving an IVP for a system of ODEs

krish

New member
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.

ZaidAlyafey

Well-known member
MHB Math Helper
What is $$\displaystyle w$$ ? ,is it a constant or a function of $t$ ?

krish

New member
What is $$\displaystyle w$$ ? ,is it a constant or a function of $t$ ?
I believe w is a constant.

ZaidAlyafey

Well-known member
MHB Math Helper
I believe w is a constant.
Aha , try differentiating one of the equations and tell me if you got any ideas .

krish

New member
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.

ZaidAlyafey

Well-known member
MHB Math Helper
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

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

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

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

krish

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

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

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?

ZaidAlyafey

Well-known member
MHB Math Helper
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:

krish

New member
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.

ZaidAlyafey

Well-known member
MHB Math Helper
It's in the review questions in my Differential Equations textbook.
Ok , tell me the name of the textbook and the page number .

krish

New member
Ok , tell me the name of the textbook and the page number .
Differential Equations (From Engineering Viewpoint) - Dr. R.C. Shah
Page 309