Differential Equations general solution

[GreenPrint]

Homework Statement

For the system of differential equations

\frac{dx}{dt}=(-3x-y)
\frac{dy}{dt}=(-2x-2y)

(a) Find the general solution.
(b) Find the solution if x(0)=1 and y(0)=2.

Homework Equations

The Attempt at a Solution

I have absolutely no clue how to do this. I have never seen a problem like this. I was wondering if someone could tell me what topic this type of problem would fall under so that way I could look up similar problems and understand the concept of how to solve these type of problems.

I know how to solve initial value problems but I have never seen a system of equations like this were both x and y are a function of t and I'm given both their derivatives. Thanks for any help.

Some ideas that pop into my head right away is that it's a system of linear equations and that I can use Gaussian reduction or something but I'm not sure.

 

[rock.freak667]

Perhaps you can find dy/dx using the chain rule and then apply a substitution such as y=vx.

 

[epenguin]

Do you know how to solve (homogeneous) equations in one variable?

Differentiate your equations and see what you can get.

(Later on maybe you'll be doing the opposite but don't worry.)

 

[GreenPrint]

Ya I do but I don't see how to apply it to this. What exactly do you mean by differentiate?

I set up the system of the system of linear differential equations and got that
x = -7/6 dx/dt + 1/4 dy/dt
y = 1/2 dx/dt -3/4 dy/dt

it didn't seem to help

 

[sunjin09]

Homework Statement

For the system of differential equations

\frac{dx}{dt}=(-3x-y)
\frac{dy}{dt}=(-2x-2y)

(a) Find the general solution.
(b) Find the solution if x(0)=1 and y(0)=2.

Homework Equations

The Attempt at a Solution

I have absolutely no clue how to do this. I have never seen a problem like this. I was wondering if someone could tell me what topic this type of problem would fall under so that way I could look up similar problems and understand the concept of how to solve these type of problems.

I know how to solve initial value problems but I have never seen a system of equations like this were both x and y are a function of t and I'm given both their derivatives. Thanks for any help.

Some ideas that pop into my head right away is that it's a system of linear equations and that I can use Gaussian reduction or something but I'm not sure.

rewrite your DE's in the vector form \frac{d}{dt}[x,y]^T=A\cdot[x,y]^T and try a solution of the form [x,y]^T=[x_0,y_0]^T\exp(\lambda t), you'll end up having a simple eigenvalue problem to solve ...

 

[epenguin]

I mean if any equation A = B is identically true for all t, then dA/dt = dB/dt is also true for all t.

So what do you get when you differentiate your

\frac{dx}{dt}=(-3x-y)
\frac{dy}{dt}=(-2x-2y)

by t?

(I should mention that underneath this is not really different from sunjin09's suggestion; you may or may not be familiar now with the formalism, but solving a 1st order linear d.e. in n variables and solving an nth order in 1 variable are equivalent and can be translated into each other.)