Math Is Fun Forum

evinda

Member

Registered: 2013-04-13

Posts: 105

Wronskian use identities !

Hello!!!
Could you help me at the following exercise?
Let

,

solutions of the differential equation y''+ay'+by=0  (a and b real constants) so that

is not constant.If

any solution of the differential equation ,use the identities of the Wronskian to show that there are constants

,

so that:

, d_{1}v_{1}'(0)+d_{2}v_{2}'(0)=f'(0)
and that each solution of the differential equation has the form:

.

Last edited by evinda (2013-12-11 06:37:39)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

Hi;

The Wronskian is

I do not see how to use v1 and v2?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Wronskian use identities !

Hi

I think evinda is referring to this:Wronskian.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

I used the one in Mathematica, they should be the same thing.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

The Wronskian is
|u(0)     v(0)|
|u'(0)    v'(0) |
But...how can I show that  there are constants

,

so that:

, d_{1}v_{1}'(0)+d_{2}v_{2}'(0)=f'(0) ??

Last edited by evinda (2013-12-11 09:27:13)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

What functions are u and v? They are not defined.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Wronskian use identities !

Well, here they are called v1 and v2 instead of u and v. Does not change much.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

What is v1 and v2?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Wronskian use identities !

They are two solutions of the differential equation for which it's true that v1/v2 is not constant.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

That would require solving the DE. Since when do you need the solve the DE to get the Wronskian?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

They are two solutions of the differential equation for which it's true that v1/v2 is not constant.

Exactly!!The Wronskian is:
| v1(0)    v2(0)  |
| v1'(0)  v2'(0)  |

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

That would require solving the DE. Since when do you need the solve the DE to get the Wronskian?

Which DE do you mean that I have to solve??

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

When I solve the DE, I get an answer that has to arbitrary constants s1 and c2

I can not eliminate them without some more information.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

That yields a Wronskian of

Now what?

I haven't understood..Do you mean that I have to take the derivative of the Wronskian and solve the DE or which DE do you mean??

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

This DE, y''+ay'+by=0

If v1 and v2 are solutions of that DE then you have to solve the DE to get the solutions. I have already done that.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

I found the Characteristic equation,that is

that has discriminant

with

So,

Am I right so far?And how can I continue?

Last edited by evinda (2013-12-11 10:10:12)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

I am not getting that as a solution to the DE. Where is c1 and c2 in your solution?

Did you copy the question correctly?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Wronskian use identities !

Hi evinda

Those are the solutions, but I don't think you are supposed to use them.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

Should not you have used the roots rather than the discriminant?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

I am not getting that as a solution to the DE. Where is c1 and c2 in your solution?

Did you copy the question correctly?

Yes,I copied the question correcty..Should I have used

  and

instead of c_{1}  and

?

Last edited by evinda (2013-12-11 10:14:50)

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

Hi evinda

Those are the solutions, but I don't think you are supposed to use them.

But???How else can I start???

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

Should not you have use the roots rather than the discriminant?

Where do you mean?At the general solution?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Wronskian use identities !

This is what I am getting for a solution to that DE:

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Wronskian use identities !

I found the Characteristic equation,that is

that has discriminant

with

So,

Am I right so far?And how can I continue?

So is she.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

evinda

Member

Registered: 2013-04-13

Posts: 105

Re: Wronskian use identities !

And how can I continue now??