dx/dy = x+y (1) dy/dt = x-y (2) x(0)=1 y(0)=1 a) differentiate (1) with respect

Question

dx/dy = x+y  (1)

dy/dt = x-y   (2)

x(0)=1

y(0)=1

a) differentiate (1) with respect to t and solve for dy/dt

b) solve (1) for y

c) use the expressions for dy/dt and y to sub into (2) to eliminate y

d) solve your result to find x(t).  you should have constraints c1 and c2 in your solution

e) repeat process to find y(t).  you should have c3 and c4 in your solution

f) you should have 4 constraints with 2 in each solution.  there are really only 2.  sub both of your solutions together with the derivative of the x solution into (1).  you should be able to get c3 and c4 in terms of c1 and c2.  rewrite your solution for y in terms of c1 and c2

g) now use the initial conditions x(0)=1, y(0)=1 to find c1 and c2 and rewrite x(t) and y(t)

Explanation / Answer

a) d^2 x/dt^2 = dx/dt + dy/dt

so dy/dt = d^2x/dt^2 - dx/dt

b) y = dx/dt - x

c) d^2 xdt^2 - dx/dt = x - dx/dt + x

d^2 x/d^2 = 2x

d) characteristic equation

c^2 - 2 = 0
c = +/- sqrt(2)

so x = c1 e^(sqrt(2) t) + c2 e^( - sqrt(2) t)

d) y = dx/dt - x = (sqrt(2) c1 - c1) e^(sqrt(2) t) + ( - sqrt(2) c2 - c2) e^(-sqrt(2) t)

g)

x(0) = 1 means c1 + c2 = 1 so c2 = 1-c1



y(0) = 1 means sqrt(2) c1 -c1 - sqrt(2) c2 - c2 = 1

sqrt(2)*c-c - sqrt(2) (1-c) - (1-c)=1

c1=1/2 + 1/sqrt(2)

so c2 = 1/2 - 1/sqrt(2)

x = (1/2 + 1/sqrt(2)) e^(sqrt(2) t) + (1/2 - 1/sqrt(2)) e^(-sqrt(2) t)
y = (1/2 + 1/sqrt(2))(sqrt(2) -1) e^(sqrt(2) t) + (-sqrt(2) -1)(1/2 - 1/sqrt(2)) e^(-sqrt(2) t)

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