Shade said:Homework Statement
Given the below stated equations I need to find the exact polynomial given the initial condition.
y(0) = 1
y = 4*t*sqrt(y)
Homework Equations
The Attempt at a Solution
I simply disregard the initial value condition and get y = t^4
How can I find the fourth order polynomial with the given initial value?
( see also a former thread https://www.physicsforums.com/showthread.php?t=111094 )
A first order ordinary differential equation (ODE) initial value problem is a type of mathematical problem that involves finding a function that satisfies a given differential equation and also satisfies a set of initial conditions, usually in the form of a starting point or a starting value for the function. The solution to the problem is a function that describes the behavior of a dependent variable over a continuous range of independent variables.
Some common techniques for solving first order ODE initial value problems include separation of variables, integrating factors, substitution methods, and using power series or numerical methods. The specific technique used depends on the form of the ODE and the initial conditions given.
Initial value problems are important in mathematical modeling because they allow us to describe and predict the behavior of a system over time. By solving the initial value problem, we can determine the values of the dependent variable at any point in the independent variable's range, which is essential for understanding and analyzing real-world systems.
No, not all first order ODEs can be solved analytically. Some ODEs have no closed form solution, meaning they cannot be expressed in terms of elementary functions. In these cases, numerical methods or approximations are used to find a solution.
First order ODE initial value problems have many real-world applications, such as modeling population growth, chemical reactions, and electrical circuits. They are also used in engineering, physics, and economics to describe and predict the behavior of various systems over time.