# Systems of odes first order linear equations

### System of differential equations solver

And that should be true for all x's, in order for this to be a solution to this differential equation. In order for this to be true for all x's. Well, in order for a constant value to be equal to a coefficient times x plus some other constant value the coefficient on X must be equal to zero. Then we could use that information, because we know that, we know that m is equal to, is equal to 3b minus 5. So here we have a differential equation. It's completely reasonable that this constant could be equal to m. Now remember, this needs to be true, this needs to be true for all x's. So dy dx. Let me find a new color here, maybe this blue. It makes sense that the number of prey present will affect the number of the predator present.

We'll just take the derivative here with respect to x dy dx is derivative of mx with respect to x, is just going to be m. So let's do that.

## Find the solution to the linear system of differential equations satisfying the initial conditions

So we could use this first one. So somehow, this thing must disappear. So assuming you have paused again and had a go at it, let's just keep algebraically manipulating this. This will lead to two differential equations that must be solved simultaneously in order to determine the population of the prey and the predator. So here we have a differential equation. So let's work through it. Let me find a new color here, maybe this blue. And just as a repeated reminder, this has to be true for all x's. Remember y is equal to mx plus b. Now remember, this needs to be true, this needs to be true for all x's. If we know that this kind of a solution can be described in this way, we need to figure out some m's and b's here. So zero must be equal to 3m minus 2, and m is equal to 3b, m is equal to 3b minus 5. So if we were to take these two and add them together that's going to be negative 2 plus 3m times x, or we could write this as 3m minus 2 times x, and then you have your constant terms, so you have these terms right over here, so plus 3b minus 5, and of course that's all going to be equal to m.

And of course derivative of b with respect to x, just a constant, so it's just going to be zero. This is telling us that if we were to take the derivative of this with respect to x, if we take the derivative of mx plus b with respect to x, that that should be equal to negative 2 times x plus 3 times y.

This will lead to two differential equations that must be solved simultaneously in order to determine the population of the prey and the predator. The whole point of this is to notice that systems of differential equations can arise quite easily from naturally occurring situations.

And that should be true for all x's, in order for this to be a solution to this differential equation. So somehow, this thing must disappear. The largest derivative anywhere in the system will be a first derivative and all unknown functions and their derivatives will only occur to the first power and will not be multiplied by other unknown functions.

Let me find a new color here, maybe this blue. Is that, did I do that right?

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