Please help. Solve

dx/dt + y = sint , x(0)=2

dy/dt + x = cost , y(0)=0

this is one problem that comes in my laplace transform chapter. Any method can be used

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- Nov 26th 2010, 05:47 PMrafi98System of DE's.
Please help. Solve

dx/dt + y = sint , x(0)=2

dy/dt + x = cost , y(0)=0

this is one problem that comes in my laplace transform chapter. Any method can be used - Nov 26th 2010, 06:09 PMProve It
From DE 1: $\displaystyle \displaystyle \frac{dx}{dt} + y = \sin{t}$

$\displaystyle \displaystyle y = -\frac{dx}{dt} + \sin{t}$

$\displaystyle \displaystyle \frac{dy}{dt} = -\frac{d^2x}{dt^2} + \cos{t}$.

Substituting into DE 2:

$\displaystyle \displaystyle -\frac{d^2x}{dt^2} + \cos{t} + x = \cos{t}$

$\displaystyle \displaystyle -\frac{d^2x}{dt^2} + x = 0$

$\displaystyle \displaystyle \frac{d^2x}{dt^2} - x = 0$.

Now, the characteristic equation is $\displaystyle \displaystyle m^2 - 1 = 0$

$\displaystyle \displaystyle m^2 = 1$

$\displaystyle \displaystyle m = \pm 1$.

So the solution to the homogeneous equation is $\displaystyle \displaystyle x = Ae^{t} + Be^{-t}$.

Follow a similar process to find $\displaystyle \displaystyle y$. - Nov 26th 2010, 06:19 PMrafi98
- Nov 26th 2010, 06:26 PMProve It
- Nov 26th 2010, 07:42 PMrafi98
What is small letter a and b . And i dont know the formula of your second reply.

- Nov 26th 2010, 07:45 PMProve It
From the second DE

$\displaystyle \displaystyle \frac{dy}{dt} + x = \cos{t}$

$\displaystyle \displaystyle x = -\frac{dy}{dt} + \cos{t}$

$\displaystyle \displaystyle \frac{dx}{dt} = -\frac{d^2y}{dt^2} - \sin{t}$.

Substitute into the first DE and solve for $\displaystyle \displaystyle y$. - Nov 27th 2010, 04:32 AMKrizalid