Solving Systems of Linear Differential Equations

• shelovesmath
In summary: I'm sorry, it was a poorly worded question. If you have a differential equation specifically of the form:a1(t)x' +a2(t)y' + a3(t)x + a4(t)y = f(t)b1(t)x' +b2(t)y' + b3(t)x + b4(t)y = g(t)which my book calls a special case of a linear system of differential equations, when specifically solving it by using the differential operator x' = Dx and y'
shelovesmath
Hi all.

Woohoo, I'm in diff eq now. Gosh, I've been on this board since Calculus I think. Sorry I usually only come around when I have a question. :\

Sooooo, Thursday my professor did an example of solving a system of linear differential equations with using the linear operator D.

There was no explanation involved, just "Here's how you solve this one."

So, now I'm doing my homework, and I need to clarify, and this might seem like a silly question, but bear with me.

Is the whole idea just to get the system into a form that is solvable only by linear methods?

For example, if you have a system with 2 equations, in 2 unknowns, say x and y, and you eliminate y, you get it into a form where you will solve for x, then rinse, lather, repeat, go back eliminate and solve for y.

If after eliminating x or y, it can be any order, right? And, if it's first order, it will be a linear or Bernoulli form, if it's second order, it will be undetermined coefficients or variation of parameters form, etc.

It seems to me that since it's a linear system, the solution should ONLY be solvable by linear means. No exact, no separable, no homogeneous method. Just linear methods.

Is this correct?

shelovesmath said:
Hi all.

Woohoo, I'm in diff eq now. Gosh, I've been on this board since Calculus I think. Sorry I usually only come around when I have a question. :\

Sooooo, Thursday my professor did an example of solving a system of linear differential equations with using the linear operator D.

There was no explanation involved, just "Here's how you solve this one."

So, now I'm doing my homework, and I need to clarify, and this might seem like a silly question, but bear with me.

Is the whole idea just to get the system into a form that is solvable only by linear methods?

For example, if you have a system with 2 equations, in 2 unknowns, say x and y, and you eliminate y, you get it into a form where you will solve for x, then rinse, lather, repeat, go back eliminate and solve for y.

If after eliminating x or y, it can be any order, right? And, if it's first order, it will be a linear or Bernoulli form, if it's second order, it will be undetermined coefficients or variation of parameters form, etc.

It seems to me that since it's a linear system, the solution should ONLY be solvable by linear means. No exact, no separable, no homogeneous method. Just linear methods.

Is this correct?

I have no idea what you mean by that last paragraph. If you have 2 linear differential equations in the two unknown functions, x and y, you can eliminate one, say y, to get a single second order linear differential equation in the other function, x.

"Exact", "separable", and "homogenous" apply to first order equations, not higher order. However, you need to be aware that, unfortunately, the term "homogeneous" is now used in a different way: a linear equation of higher order is said to be "homogeneous" if every term involves the dependent function, y, or one of its derivatives. That has nothing to do with the concept of "homogeneous" for first order equations.

HallsofIvy said:
I have no idea what you mean by that last paragraph. If you have 2 linear differential equations in the two unknown functions, x and y, you can eliminate one, say y, to get a single second order linear differential equation in the other function, x.

"Exact", "separable", and "homogenous" apply to first order equations, not higher order. However, you need to be aware that, unfortunately, the term "homogeneous" is now used in a different way: a linear equation of higher order is said to be "homogeneous" if every term involves the dependent function, y, or one of its derivatives. That has nothing to do with the concept of "homogeneous" for first order equations.

I'm sorry, it was a poorly worded question.
If I have a differential equation specifically of the form:

a1(t)x' +a2(t)y' + a3(t)x + a4(t)y = f(t)
b1(t)x' +b2(t)y' + b3(t)x + b4(t)y = g(t)

which my book calls a special case of a linear system of differential equations, when specifically solving it by using the differential operator x' = Dx and y' = Dy, the goal is to use the elimination method to isolate x, find a general solution, then use elimination method again to find y, find another general solution, and in the end, make sure the number of scalars are equal to the order of the differential equations you got after using elimination.

What I'm curious about are those differential equations you get after doing the elimination. If you end up with a first order differential equation, will it always be linear or bernoulli? Is it even possible to end up with a first order differential equation after using the elimination method?

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 the derivatives of one or more variables. The equations are linear, meaning that the highest power of the variable in each equation is 1. These systems are used to model and solve problems in various fields, including physics, engineering, and economics.

2. How do you solve a system of linear differential equations?

There are several methods for solving a system of linear differential equations, including substitution, elimination, and matrix methods. The most common approach is to use a technique called the "elimination method," where you manipulate the equations to eliminate one variable at a time until you are left with a single equation that can be solved for the remaining variable.

3. What is the importance of solving systems of linear differential equations?

Solving systems of linear differential equations is important because it allows us to model and analyze complex systems in various fields. These equations can help us understand how systems change over time and make predictions about their future behavior. They are also essential in engineering and physics, where they are used to design and control systems such as circuits and mechanical systems.

4. What are some real-world applications of solving systems of linear differential equations?

Systems of linear differential equations have many real-world applications, including modeling population growth, analyzing chemical reactions, and predicting the motion of objects under the influence of forces. They are also used in economics to study the behavior of markets and in engineering to design and control systems.

5. Are there any limitations to solving systems of linear differential equations?

While systems of linear differential equations are powerful tools for modeling and analyzing complex systems, they do have some limitations. For example, they may not accurately represent nonlinear systems, and they can become very complex and difficult to solve for larger systems. In some cases, numerical methods may be necessary to approximate a solution. Additionally, the initial conditions and assumptions used in the model may affect the accuracy of the solution.

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