A solution for a system of differential equations?

Question

I want to check answer for specific ODE solvers, for instances, solving:

$x_1' = 1/5\; x_1 + 4/5\; x_2$

$x_2' = 4/5\;x_1 + 1/5\; x_2$

$x_1(0) = 1$, $x_2(0) = 3$

I've just learnt how to solve these types of equations and it'd be helpful if I could check my answer.

I got an answer of $x_1 = 2e^t -e^{-3/5t}$ and $x_2 = 2e^t + e^{-3/5t}$

Answer 1

Grapher seems to agree, the picture is easy to examine if you just take it down from the page.

Answer 2

Your initial value problem (system of ODEs + initial conditions) enjoys global uniqueness, and hence once you find a solution (regardless how you did) and you have verified that this pair of functions satisfies the equations and initial conditions then you are done.

Your solution is THE solution of the IVP.

Answer 3

If I were you, I would do the system as follows:

$$x'=1/5x+4/5y,~~y'=4/5x+1/5y\Longrightarrow Dx=1/5x+4/5y,~~ Dy=4/5x+1/5y$$ where I employ $D$ instead of ' . Now you have:

$$(D-1/5)x-4/5y=0,~~(D-1/5)y-4/5x=0$$ Solving the latter system as you always do in pre-calculus, you have $$(D-1/5)^2y-16/25y=0,~~(D-1/5)^2x-16/25x=0$$ Both ones are easy ODEs of second orders with constant coefficients.