A question about differential equations

[Nikitin]

Hi,I'm new to these and thus my question might sound stupid: Do differential equations ALWAYS have just one or zero general solutions? I know each diff.equation can have multiple particular solutions, but can it only have one or zero general solutions?

 

[hunt_mat]

Depending on the DE (if it is linear), you can 2 solutions, add them together and obtain another solution. I think your question relates to either initial conditions or boundary conditions which will lead to uniqueness.

 

[HallsofIvy]

That depends upon what you mean by "general solution".

An example used in many texts is $y'= y^{1/2}$. That's easily separable so we get $y^{-1/2}dy= dx$ and, integrating, $2y^{1/2}= x+ C$ or $y= (x+ C)^2/4$. However, it is clear that y(x)= 0, for all x, also satisfies that differential equation. That means that, for y(1)= 0, for example, we can $y= (1+ C)^2/4= 0$ so that C= -1. So that both $y= (x- 1)^2/4$ and y= 0 for all x satisfy both the differential equation and the initial condition.

I think you need to look at the concepts of "existence and uniqueness" for initial value problems which is probably given in your textbook.