# Existence and uniqueness of differential solution, help?

by Lengalicious
Tags: differential, existence, solution, uniqueness
 P: 127 Ok so ill give an example, x'(t) = log(3t(x(t)-2)) is differential equation where t0 = 3 and x0 = 5 The initial value problem is x(t0) = x0. So what i'de do is plug into initial value problem to get x(3) = 5, so on a graph this plot would be at (5,3)? Then plop conditions into differential equation so: x'(3) = log(3*3(5 - 2)) = 1.43 which would be at a plot (1.43,3)? So x'(3) < x(3), does this mean that the solution is unique for all t? If so, why is this? Just want to understand 100%.
 PF Patron Sci Advisor Thanks Emeritus P: 38,416 I am confused as to what you are saying. "Ok so ill give an example, x'(t) = log(3t(x(t)-2)) is solution to equation where t0 = 3 and x0 = 5". x'(t)= ... is NOT a "solution" to a differential equation, it is a differential equation. Or do you mean to say that x(t)= log(4t(x(t)- 2)) is a solution to some unknown differential equation? Then whether or not that equation is unique depends upon exactly what the differential equation is.
 P: 127 Ok, yes my mistake its a differential equation i guess? This is where I myself am confused =/. If its a differential equation then how do you find out whether it has a unique solution or not?
P: 187

## Existence and uniqueness of differential solution, help?

If dx/dt=f(x,t), and both f and the partial of f w.r.t. x are continuous in some region about your initial point then then exists a unique solution of the initial value problem in some region containing that initial point. This is a local property. The proof is dependent upon finding a sequence of solutions via Picard iteration which converges to the unique solution. You can find this proof in any introductory ode text.
PF Patron
P: 1,942
 Quote by Lengalicious Ok so ill give an example, x'(t) = log(3t(x(t)-2)) is differential equation where t0 = 3 and x0 = 5 The initial value problem is x(t0) = x0. So what i'de do is plug into initial value problem to get x(3) = 5, so on a graph this plot would be at (5,3)? Then plop conditions into differential equation so: x'(3) = log(3*3(5 - 2)) = 1.43 which would be at a plot (1.43,3)? So x'(3) < x(3), does this mean that the solution is unique for all t? If so, why is this? Just want to understand 100%.
You must interpret x'(3) as the slope of the tangent at your first point, so you'd draw
x(3+s) approx x(3)+x'(3)s = 5+1.43 s for small s. Then you continue from that point. This gives you an idea of the solution, not very accurate though. It is called Euler's method, but it can be refined to give an existence and uniqueness proof, if you make appropriate assumption about the differential equation.
 P: 127 Ok, I sort of get it but still slightly confused, an exact question I have is: In the following case: x'(t) = log(3t(x(t)-2)), where t0 = 3 and x0 = 5 Does the theorem of existence and unicity guarantee an existence of a solution for the initial value problem x(t0) = x0? Justify your answer. What would your answer be to this? Would help me understand if I got a model answer.
PF Patron