More than N Solutions to Nth Order DE?

[cocopops12]

like for example if we had a 2nd order differential equation
is it possible that we can have more than 2 non-trivial solutions to it?

same question applies to 1st order,.......Nth order

is it always the case that we get the number of solutions same as the order of the equation?

thanks!

 

[theorem4.5.9]

This is true if the differential equation is linear. In this case you are looking for solutions of the nullspace, which whose dimension equals the order of the ODE. It's really a neat application of linear algebra.

 

[Vargo]

Hi cocopops.

Well, in general the solution to an nth order differential equation is an n-parameter family of functions. For example, y'=y, the solution is y=ce^x. c is the 1 parameter.

If y''+y=0, then y= a cos(x) + b sin(x), a,b are the parameters. There are an infinite number of solutions but you can specify one of them with two numbers.

If you are talking about linear equations, then an nth order equation will have solutions "generated" by n independents solutions. Like sin(x), cos(x) in the above example. But this feature is special to linear equations and is not true for other equations.

There are other technicalities. For example y'=y^(1/3) has the one parameter family of solutions
y= [(2/3)( x - a)]^(3/2)
But it also has y=0 as an "extra" solution. (more precisely, when the initial value of the function is y=0, then there is more than one solution).

 

[AlephZero]

Yes, some differential equations have "singular solutions" that don't fit the general pattern of famiilies of solutions containing arbitary constants.

The basic requirement for these to exist is that the DE does not specify unique derivative(s) at every point $x_1, x_2, \dots x_n, y$. A common reason for this is that some functions in the DE are zero for some particular values.

Take for example an equation of the form f(x,y)dy/dx + g(x,y) = 0. If there is a point (x,y) where f(x,y) = 0 and g(x,y) = 0, at that point dy/dx can take any value and there is the possibility that singular solutiosn exist passing through that point.

http://en.wikipedia.org/wiki/Singular_solution