mistereko said:Thanks very much. That should be ok. :)
mistereko said:So, will I have two solutions in the end?
mistereko said:Homework Statement
I just need to know how to start this. I've never seen a piece wise ODE before and I don't really know where to start to be honest. I know it's non-linear and it's of order one.
dy/dx = (1/3)y - 3, y > 0 and dy/dx = -(1/3)y - 3 ≤ 0. y(0) = 1 with y(x) [itex]\in[/itex] C0 [0,[itex]\infty[/itex])
I'm trying to find the non-trivial solution.
Homework Equations
The Attempt at a Solution
Ray Vickson said:It is important to know if you mean that
[tex] y' = \frac{y}{3} - 3, \; y > 0, \text{ and } y' = -\frac{y}{3} - 3, \;
y \leq 0, [/tex]
or
[tex] y' = \frac{y}{3} - 3, \; x > 0, \text{ and } y' = -\frac{y}{3} -3, \; x \leq 0.[/tex]
Both systems have [itex]C^0[/itex] solutions; one of them has [itex]C^1[/itex] solutions, but the other does not (except for one very particular choice of initial conditions).
RGV
A non-linear ODE with IV (initial value) problem is a type of differential equation that involves a non-linear relationship between the dependent variable and its derivatives, along with a set of initial conditions. These initial conditions specify the values of the dependent variable and its derivatives at a given point in the independent variable.
A linear ODE (ordinary differential equation) is one where the dependent variable and its derivatives have a linear relationship. This means that the dependent variable and its derivatives appear only to the first power. In contrast, a non-linear ODE with IV problem has a non-linear relationship between the dependent variable and its derivatives, meaning they can appear to higher powers.
Non-linear ODE with IV problems have a wide range of applications in various fields, including physics, engineering, economics, and biology. Some examples include modeling population growth, predicting the spread of diseases, analyzing chemical reactions, and understanding the behavior of complex systems such as weather patterns.
Unlike linear ODEs, there is no general method for solving non-linear ODE with IV problems. However, there are various techniques and numerical methods that can be used to approximate solutions, such as the Euler method, Runge-Kutta methods, and series solutions.
One of the main challenges of solving non-linear ODE with IV problems is that they do not have a general solution, making it difficult to find an exact solution. Additionally, non-linear ODEs often have complex and unpredictable behavior, making it challenging to analyze and interpret their solutions. This requires advanced mathematical techniques and computational tools to approximate solutions and understand their behavior.