# Solving Systems of Linear Differential Equations

• shelovesmath
In summary, the conversation discusses the process of solving a linear first order differential equation system involving x and y. The first step is to eliminate x and get y alone, followed by finding an integrating factor and solving the linear ODE. The second step is to eliminate y and get x alone, using a similar process as before. The conversation also notes the importance of using correct notation and recommends writing the equations in matrix notation.
shelovesmath

1. x' + y' - x = -2t
x' + y' - 3x -y = t^2

2. x' = Dx, y' = Dy

3.
I eliminated x to get y alone by multiplying the first row by (D-3) and the second row by (D-1)

(D-1)(D-3)x + (D-3)Dy = -2t(D-3)
(D-1)(D-3)x + (D-1)(D-1) = t^2(D-1)

then subtracted to get: (D+1)y = -t^2 -4t + 2

I put this into a differential equation form y' + y = -t^2 - 4t + 2 This is linear first order, and my integrating factor is e^t (at this point I will solve this linear ODE)

For getting x by itself, I eliminated y by multiplying the first row by (D-1) and the second row by D

(D-1)(D-1)x + D(D-1)y = -2t(D-1)
D(D-3)x + D(D-1)y = Dt^2

then subtracted to get: (D+1)y = -2

I put this into a differential equation form y' + y = -2 This is linear first order, and my integrating factor is also e^t (at this point I will solve this linear ODE)

I just need to know that I'm good up to here.

First: your notation is a bit confusing. Also derivative operators (usually) act on the right, so you multiply them from the left, and therefore have -2 (D-3) t instead of -2 t (D-3). However, your result is correct, that is
y' + y = -t^2 - 4t + 2. (the integrating factor e^t is also correct)

Afterwards (in the second calculation) I think you confused y and x, since I get
x' + x = -2.

I recommend writing it in matrix notation.

Antiphon said:
I recommend writing it in matrix notation.

I have to use this method specifically for my homework assignment.

## 1. What is a system of linear differential equations?

A system of linear differential equations is a set of two or more equations that involve derivatives of an unknown function. These equations are linear because they are of the first degree in the unknown function and its derivatives. They can be solved simultaneously to find the unknown function.

## 2. Why is it important to solve systems of linear differential equations?

Solving systems of linear differential equations is important in many fields of science and engineering, such as physics, chemistry, and economics. These equations can model real-world situations and help us understand and predict the behavior of systems over time. Solving them allows us to make informed decisions and develop solutions to complex problems.

## 3. What methods are used to solve systems of linear differential equations?

There are several methods for solving systems of linear differential equations, including substitution, elimination, and using matrices. Each method has its advantages and may be more suitable for certain types of equations. Computer software and calculators can also be used to solve systems of linear differential equations.

## 4. Can systems of linear differential equations have multiple solutions?

Yes, a system of linear differential equations can have multiple solutions. This means that there can be more than one function that satisfies all of the equations in the system. In some cases, the solutions may be unique, while in others there may be an infinite number of solutions.

## 5. How can I check if my solution to a system of linear differential equations is correct?

To check if a solution to a system of linear differential equations is correct, you can substitute the solution into each equation and see if it satisfies all of them. If it does, then the solution is correct. Alternatively, you can use computer software or calculators to verify the solution.

Replies
3
Views
845
Replies
3
Views
1K
Replies
7
Views
850
Replies
1
Views
786
Replies
6
Views
1K
Replies
3
Views
578
Replies
10
Views
1K
Replies
3
Views
1K
Replies
10
Views
1K
Replies
4
Views
1K