Is This Proof for Finding a Third Independent Solution to a 3rd Order ODE Valid?

[AdrianZ]

Here's the problem:
Show that if we have two independent solutions to a 3rd order homogenous linear differential equation y'''+a₂y''+a₁y'+a₀=0 we can always find a third solution which is independent from the two given solutions.

Here is my proof:
since y₂ and y₁ are linearly independent, there exists a non-constant function u such that y₂=uy₁.
Now, If we plug y₂ into the equation, we can reduce the order of the original equation to a 2nd order linear ODE of the form a(x)v+b(x)v'+c(x)'' = 0 where:
$$u=y_2/y_1=\int{v(x)dx}$$

Now, If I set v=wy₃, and I suppose t=w', then doing the same process I can reduce the ODE a(x)v+b(x)v'+c(x)v''=0 into a first order linear ODE of the form r(x)t+s(x)t'=0.
This ODE is always solvable and the solution to this ODE is:
$$t=Ce^{\int{-r(x)/s(x)dx}}$$
so, w would be:
$$w=C\int{e^{\int{-r(x)/s(x)dx}}}$$

Having w and v, we can find y₃=v/w.

This shows us that y₃ can be found, but I don't know how to show that y₃ is necessarily independent from y₁ and y₂.

Any idea that completes my proof would be highly appreciated.

 

[Charles49]

Did you try using the wronskian?

 

[AdrianZ]

I used Wronskian of y₁ and y₂ to prove that v is not a constant function, but I don't know how to use Wronskian to prove that y₃ is linearly independent from y₁ and y₂. You mean that I should form a 3 by 3 determinant that each row is a vector whose components are y_i y_i' and y_i'' (i=1,2,3)? then I'll have to show that the Wronskian isn't equal to 0? Is that what you meant?

 

[Charles49]

Yeah, you have to take a 3x3 matrix determinant and show it's not zero.

 

[AdrianZ]

but How could I form that determinant? I mean I don't precisely have y₃, in case of y₁ and y₂ I got some information that they are linearly independent, but regarding y₃ I got no information explained precisely. would you give me a hint?

 

[Charles49]

You just gave me y3=y1/y2.

Just use quotient rule twice on y3 and form the wronskian.

See http://en.wikipedia.org/wiki/Wronskian

You want to know how to take 3x3 determinant, use the Sarrus rule.

Am I misunderstanding?

 

[Charles49]

If you have an expression for y3, all the hard work is done.

 

[AdrianZ]

You just gave me y3=y1/y2.

Just use quotient rule twice on y3 and form the wronskian.

See http://en.wikipedia.org/wiki/Wronskian

You want to know how to take 3x3 determinant, use the Sarrus rule.

Am I misunderstanding?

I hope I haven't done such a mistake in typing my proof, because y3 isn't necessarily equal to y1/y2, u = y1/y2, not y3.
y3=v/w. but v is a solution of the 2nd order linear ODE, so I don't precisely have it.

If you have an expression for y3, all the hard work is done.

I doubt I could find a precise expression for y3.

 

[Charles49]

I think there is a different way of doing this problem. Construct the characteristic equation:
$$z^3+a_2z^2+a_1z+a_0.$$
Since this is a cubic, there are 3 types of zeros (by the fundamental theorem of algebra):

1. 3 distinct real zeros
2. 2 repeated and 1 distinct real zeros
3. 1 real zero and two conjugate pairs of zeros

Observe that there cannot be three complex roots because complex roots come in pairs.

Now show that in each case, if there are two linearly independent zeros, then the third has to be linearly independent.

 

[jackmell]

Here's the problem:
Show that if we have two independent solutions to a 3rd order homogenous linear differential equation y'''+a₂y''+a₁y'+a₀=0 we can always find a third solution which is independent from the two given solutions.

Here is my proof:
since y₂ and y₁ are linearly independent, there exists a non-constant function u such that y₂=uy₁.
Now, If we plug y₂ into the equation, we can reduce the order of the original equation to a 2nd order linear ODE of the form a(x)v+b(x)v'+c(x)'' = 0 where:
$$u=y_2/y_1=\int{v(x)dx}$$

Now, If I set v=wy₃, and I suppose t=w', then doing the same process I can reduce the ODE a(x)v+b(x)v'+c(x)v''=0 into a first order linear ODE of the form r(x)t+s(x)t'=0.
This ODE is always solvable and the solution to this ODE is:
$$t=Ce^{\int{-r(x)/s(x)dx}}$$
so, w would be:
$$w=C\int{e^{\int{-r(x)/s(x)dx}}}$$

Having w and v, we can find y₃=v/w.

This shows us that y₃ can be found, but I don't know how to show that y₃ is necessarily independent from y₁ and y₂.

Any idea that completes my proof would be highly appreciated.

I believe you have insufficiently shown how to go from a second-order to a first-order equation. Also, when you get something you think is right, would be nice if you could check it with a real-live problem that you know the answer to and you can do that check numerically. That is, get two solutions, then numerically compute whatever difficult expression for the third and then see if it matches (numerically at least), what you know is correct. For example, does your expression above for y_3 agree with an actual set of solutions for a third-order ODE?