Form of Solution for First Order ODE T'(t) - (1 - n^2/4)T(t) = 0

In summary, the conversation discusses different forms of solutions for various differential equations. There is a question about the form for the equation y'(t) - ay(t) = 0 and a discussion about the solutions for the equations T'(t) - (1 - \frac{n^2}{4})T(t) = 0 and y' = (y - x)^2. It is suggested to try a substitution of v=y-x to determine the form of the solution for the last equation.
  • #1
leopard
125
0
y'(t) - ay(t) = 0

What is the form of the solution? [tex]C \cdot e^{at}[/tex]

?I have this ODE:

[tex] T'(t) - (1 - \frac{n^2}{4})T(t) = 0[/tex]

If I'm right, the solutions should be of the form

[tex]C \cdot e^{(1- \frac{n^2}{4})t}[/tex]

My book, however, says [tex]C \cdot e^ {1- \frac{n^2}{4}t}[/tex]

Who's right?
 
Last edited:
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  • #2
I think the book forgot some parentheses
 
  • #3
Brilliant.

And how about the equation

y' = (y - x)^2

what's the form of the solution here?

I find it hard to determine the form of solution of differential equations.
 
Last edited:
  • #4
I would try a simple substitution first. How about v=y-x? Now see if you can separate it in those variables.
 

1. What is a first order ODE?

A first order ODE (ordinary differential equation) is an equation that involves an unknown function and its first derivative. It can be written in the form of dy/dx = f(x,y), where y is the unknown function and f(x,y) is some given function. The goal is to find a solution for y that satisfies the equation for all values of x.

2. What is the form of solution for a first order ODE?

The general form of solution for a first order ODE is y = y0 * eF(x), where y0 is the initial condition (the value of y at a specific point) and F(x) is the antiderivative of f(x,y).

3. How do you solve a first order ODE?

There are various methods for solving first order ODEs, including separation of variables, integrating factors, exact equations, and substitution. In this particular equation, we can use the method of separation of variables to find the solution.

4. What is the meaning of the parameters in the given equation?

The parameter n represents a constant in the equation that affects the behavior of the solution. The value of n can change the type of solution (e.g. exponential, trigonometric) and the stability of the solution. In this case, the value of n also affects the rate of growth or decay of the solution.

5. How does this equation relate to real-world problems?

The form of solution for this first order ODE can be applied to various real-world problems, such as population growth, radioactive decay, and electrical circuits. It can also be used to model the behavior of physical systems in physics and engineering. By solving the equation, we can understand the behavior and predict the future state of these systems.

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