Can an Exponential Factor Create a New Solution for a Second Order ODE?

  • Thread starter Thread starter intervoxel
  • Start date Start date
  • Tags Tags
    Ode
intervoxel
Messages
192
Reaction score
1
Is it possible to form a new solution of a second order ODE by multiplying it by an exponential factor?
 
Physics news on Phys.org
Not clear what you mean. In a second-order LINEAR diff eq, if you have one solution u(x) you can find the general solution by trying the form f(x)u(x) ... This should be in standard ODE textbooks.
 
What I mean is that I have two particular solutions even and odd that diverge at infinity, but I noted that if I multiply them by exp(-z^2 / 4) they behave properly. I'm trying to justify this procedure. Substituting the product of each back into the differential equation doesn't seem to work.
 
In general, no. Multiplying a solution to a d.e. by another function, exponential or not, does NOT give a new solution.
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
View full post »

Similar threads

I Second Order ODE with Exponential Coefficients
Replies
1
Views
2K
I When is an ODE (numerically) reversible in time?
Replies
4
Views
2K
I Converting Second Order ODE to Hypergeometric Function
Replies
7
Views
3K
I How to do Magnus expansion for time-dependent companion matrix?
Replies
7
Views
2K
I Can you check the solution for this second order ODE?
Replies
6
Views
2K
A Existence of unique solutions to a first order ODE on this interval
Replies
2
Views
2K
A Spectral Theorem to Convert PDE into ODE
Replies
2
Views
3K
Solving Coupled First Order ODEs: Is There a Closed Form Solution?
Replies
28
Views
3K
MHB Solutions of the ODEs - 2 first order linear equations
Replies
6
Views
1K
MHB Converting a Second-Order IVP into a System of Equations: Can Substitution Help?
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top