What is the derivative of x(t,a) with respect to a when t=1 and a=0?

In summary, the student is trying to find a solution to an equation involving a and x, but is having trouble doing so. The student has tried different methods involving e, sin, and cos, but has not been successful.
  • #1
mkerikss
18
0

Homework Statement


We consider the solution to the differential equation x'(t)=-x(t)+atx(t)2, x(0)=e as a function of the variable a. Define d/da x(t,a) t=1, a=0

Homework Equations





The Attempt at a Solution



I suppose the differentiation won't be too hard, but my problem is I just don't get a solution xt,a) to the equation. I've tried splitting x'(t) into dx/dt, but that didn't work, and in desperation I've tried a number of random (ok they are not random, because I have still given it some thouht but I haven't used any special method) functions involving e, sin or cos. This is actually the part of the course that's dealing with systems of differential equation, so I've forgotten some of the stuff we learned about this type of equations about a year ago. I hope you can help!
 
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  • #2
Same as my previous post :tongue2:
I brought this up so it wouldn't get lost in the depths of the forum!
 
  • #3
mkerikss said:

Homework Statement


We consider the solution to the differential equation x'(t)=-x(t)+atx(t)2, x(0)=e as a function of the variable a. Define d/da x(t,a) t=1, a=0
There are a couple of things there that are confusing.
1. Is x a function of one variable or two? In the equation above you have x(t) and x'(t), which suggests that x is a function of one variable, t. Elsewhere you have x(t, a), which suggests that x is a function of two variables.
2. Are you supposed to find the partial of x(t, a) with respect to a, evaluated at t = 1 and a = 0? The use of the word "define" is throwing me off. Usually when "define" is used, it will give the definition of the thing being defined.
mkerikss said:

Homework Equations





The Attempt at a Solution



I suppose the differentiation won't be too hard, but my problem is I just don't get a solution xt,a) to the equation. I've tried splitting x'(t) into dx/dt, but that didn't work, and in desperation I've tried a number of random (ok they are not random, because I have still given it some thouht but I haven't used any special method) functions involving e, sin or cos. This is actually the part of the course that's dealing with systems of differential equation, so I've forgotten some of the stuff we learned about this type of equations about a year ago. I hope you can help!
 
  • #4
1. I asked myself the same question. My guess it that x is a 1-variable function, and x(t,a) is used because a is supposed to be a variable in the second part of the problem, even if it's not originally a variable of the function x. So they're sort of trying to make things clearer by using incorrect mathematic language or something. At least that's my take on the situation.

2. You understood correctly. I don't use this terminology in English very often, so sorry for that mistake :smile:
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It describes how a quantity changes over time or space, and is used to model a wide range of natural phenomena in science and engineering.

2. What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation (ODE) involves a single independent variable, while a partial differential equation (PDE) involves multiple independent variables. ODEs describe systems that change in one direction, while PDEs describe systems that change in multiple directions simultaneously.

3. How are differential equations used in science?

Differential equations are used to model and understand a wide range of natural phenomena, including population growth, chemical reactions, motion of objects, and heat transfer. They also play a crucial role in fields such as physics, chemistry, biology, and engineering.

4. What are the different methods for solving differential equations?

There are several methods for solving differential equations, including analytical methods (such as separation of variables and variation of parameters), numerical methods (such as Euler's method and Runge-Kutta methods), and computer simulations (such as finite difference and finite element methods).

5. How do differential equations relate to real-life problems?

Differential equations are used to model real-life problems because they can accurately describe the relationships between variables and their rates of change. By solving these equations, scientists and engineers can make predictions and understand the behavior of complex systems in the real world.

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