1. System of DE's.

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

2. 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 $\displaystyle \displaystyle y$.

3. Originally Posted by Prove 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 $\displaystyle \displaystyle y$.
But if we do like this , we may not be able to make use of the condition given in the question. Can i ?. Please respond. Thank you

4. Originally Posted by rafi98
But if we do like this , we may not be able to make use of the condition given in the question. Can i ?. Please respond. Thank you
Find $\displaystyle \displaystyle y$ first, see what you get.

5. What is small letter a and b . And i dont know the formula of your second reply.

6. Originally Posted by rafi98
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
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 $\displaystyle \displaystyle y$.

7. Originally Posted by rafi98
$\displaystyle s\widehat x(s)-x(0)+\widehat y(s)=\dfrac1{s^2+1}$ and $\displaystyle s\widehat y(s)-y(0)+\widehat x(s)=\dfrac s{s^2+1},$ the rest is algebra and a matter of finding inverse transforms.