Coupled first order differential equation.

[hangainlover]

Homework Statement

[itex]\frac{dx}{dt}=\gamma y[/itex]
[itex]\frac{dy}{dt}=-\gamma x[/itex]

solve for x and y

Homework Equations

The Attempt at a Solution

I know how to solve it by substitution(without using matrix)
I know how to solve a coupled second order differential equations in matrix form, but not when I am given two first order differential equations...
(I solved it by substituting y=(1\γ)dx/dt into the second equation ...
Id like to know how to solve it in matrix form

 

[mighty2000]

)

Hello there,

Thank you for your question. Solving coupled first order differential equations in matrix form is a useful technique that can help simplify the solution process. Here's how you can apply it to the given equations:

1. Write the equations in matrix form:
[d/dt(x) d/dt(y)] = [0 γ -γ 0] [x y]

2. Rewrite the matrix in terms of its eigenvalues and eigenvectors:
[0 γ -γ 0] = [iλ1 -iλ1 0 0] [u1 u2]

where λ1 = iγ and u1 = [1 i] and u2 = [1 -i].

3. Substitute the new matrix into the original equation:
[d/dt(x) d/dt(y)] = [iλ1 -iλ1 0 0] [u1 u2] [x y]

4. Simplify the equation:
d/dt[u1x + u2y] = iλ1[u1x + u2y]

5. Integrate both sides:
u1x + u2y = Ce^(iλ1t)

6. Substitute back the values of u1 and u2:
x + iy = Ce^(iλ1t)

7. Take the real part of the equation:
x = Ccos(γt)

8. Substitute this value for x into the original equation for y:
dy/dt = -γCsin(γt)

9. Integrate both sides:
y = -Ccos(γt) + D

10. Substitute the values of x and y into the original equations:
dx/dt = γ(-Ccos(γt) + D)
dy/dt = -γCcos(γt) - γD

11. Simplify and solve for C and D using initial conditions if given.

I hope this helps you understand how to solve coupled first order differential equations in matrix form. Let me know if you have any further questions.