How can I solve this differental equation?
I have this as a part of my homework, and there were like 20 other differential equations that I easily solved, but this one stood out.
This is not linear, not separable. I don't know how to approach this problem. If you could tell me to what class this kind of equation belongs, and some methods to solve them, I would be very very glad.
How am I supposed to find the solution to the non-homogeneous ode ?
Heat transfer between two fluids through a sandwiched solid (coupled problem):
Two fluids flow opposite to each other on either side of a solid (T), while exchanging heat among themselves. In such a scenario, the conduction in the solid is governed by:
1)
with boundary condition as
The fluids are governed by the following equations:
2)
3)
The hot fluid initiates at and the cold fluid starts from .
The boundary conditions are and
Equation (1), (2) and (3) form a coupled system of ordinary differential equations.
It is pretty evident that using (2) and (3), Equation (1) can be re-written as:
4)
However, I have not been able to proceed further.
Some parameter values are
Consider an ODE system
,
having a candidate Lypunov function, which satisfies , and .
How to show that is compact?
How can I handle to solve 1st order linear DE?
RHS and LHS are same.
Taking log.
Then what about ln|x| ?
This problem is a sub-problem of below ODE.
My thoughts are as below.
I got the following general solution as I forcefully assumed
Continuous function composed with itself is equal to propagation of a differential equation.
This question has been bugging me for a while. It was given as the last question of a first year undergrad analysis exam and so should be solvable with little machinery, yet it seems to point straight at ODEs which have yet to be covered. Here is the question:
Let be a bounded function and a family of continuous functions with such that
Show that there exists a continuous function such that .
I have attempted this with a classmate and we've come to various levels of understanding of the question, but the required conclusion keeps escaping us. Our main issues with the question are that we have very little understanding on continuous functions composed with themselves, and additionally we haven't managed to use the and boundedness condition on F.
Note that the question also states that we are allowed to assume that the family of functions is uniquely determined by the given conditions: this again points to the domain of differential equations, which ideally we should not need to refer to in order to solve this.
How a solving a nonlinear PDE with Self-similar Solution?
I want to solve the problem :
We search for a self-similarity solution, the general form of which is as follows
from which we obtain
Now, I am very confused on how to solve the above equation and find the exact solution of , thereby finding the exact solution of . So my question is, does anyone know how to solve this ordinary differential equation?