system of differential equations

• May 14th 2008, 08:16 AM
johnbarkwith
system of differential equations
Does anybody know how to solve the system of differential equations,

y'(t) = -ay(t) + ax(t)

x'(t) = ay(t) -ax(t)

with initial conditions, y(0)=1 and x(0)=0 where a is a real constant and also with y(t) + x(t) = 1 for all t
• May 14th 2008, 08:40 AM
flyingsquirrel
Hi

Let's use the equations we're given : substitute $x(t)=1-y(t)$ in $y'(t)=-ay(t)+ax(t)$ :

$y'(t)=-ay(t)+a(1-y(t))=-2ay(t)+a$

Hence $y$ is given by the ODE $y'(t)+2ay(t)=a$ which you can solve. Once you know $y$, you can use $x(t)=1-y(t)$ to find $x$.