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

[mkerikss]

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!

 

[mkerikss]

Same as my previous post
I brought this up so it wouldn't get lost in the depths of the forum!

 

[Mark44]

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.

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!

 

[mkerikss]

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