Differential Equations general solution

In summary, the conversation discusses a system of linear differential equations and how to find the general solution and a specific solution with initial values. Methods such as Gaussian reduction and substitution are suggested, as well as rewriting the equations in vector form and solving an eigenvalue problem. The idea of differentiating the equations to get a simpler problem is also mentioned.
  • #1
GreenPrint
1,196
0

Homework Statement



For the system of differential equations

[itex]\frac{dx}{dt}=(-3x-y)[/itex]
[itex]\frac{dy}{dt}=(-2x-2y)[/itex]

(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.
 
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  • #2
Perhaps you can find dy/dx using the chain rule and then apply a substitution such as y=vx.
 
  • #3
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.)
 
  • #4
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
 
  • #5
GreenPrint said:

Homework Statement



For the system of differential equations

[itex]\frac{dx}{dt}=(-3x-y)[/itex]
[itex]\frac{dy}{dt}=(-2x-2y)[/itex]

(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 [itex]\frac{d}{dt}[x,y]^T=A\cdot[x,y]^T[/itex] and try a solution of the form [itex][x,y]^T=[x_0,y_0]^T\exp(\lambda t)[/itex], you'll end up having a simple eigenvalue problem to solve ...
 
  • #6
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

[itex]\frac{dx}{dt}=(-3x-y)[/itex]
[itex]\frac{dy}{dt}=(-2x-2y)[/itex]

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.)
 

FAQ: Differential Equations general solution

What is a general solution for a differential equation?

A general solution for a differential equation is an equation that contains a set of solutions that satisfy the given differential equation. This means that any solution that satisfies the differential equation can be obtained as a special case of the general solution.

How do you find the general solution for a differential equation?

To find the general solution for a differential equation, you must first solve the equation by finding the antiderivative of the given function. This will give you the general solution in the form of a constant, which can then be solved for by plugging in initial conditions or boundary values.

Can a general solution be unique?

No, a general solution is not unique. This is because there are usually multiple solutions that satisfy a given differential equation. The general solution accounts for all possible solutions, and it is up to the initial conditions or boundary values to determine the specific solution.

What is the difference between a general solution and a particular solution?

A general solution is an equation that contains a set of solutions that satisfy the given differential equation. A particular solution, on the other hand, is a specific solution that satisfies the given differential equation and also meets certain initial conditions or boundary values.

Can a differential equation have an infinite number of solutions?

Yes, a differential equation can have an infinite number of solutions. This is because the general solution accounts for all possible solutions, and there can be multiple values for the constant that satisfy the equation. However, a particular solution will only have one unique solution that satisfies the given initial conditions or boundary values.

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