# Solutions of the ODEs - 2 first order linear equations

• MHB
• Julio1
In summary, the two equations are solved to find that X(t) is equal to a function that takes the form of a matrix equation with two exponents.
Julio1
Find the general solution of the ODE:

$\check{X_1}=X_1$

$\check{X_2}=aX_2$

where $a$ is a constant.

I find this hard to read. Is that symbol above the "X"s a double dot, the second derivative symbol? I will assume that it is.

You actually have two differential equations not one. And they are completely separate so you can solve them separately.

My question is "where did you get these?" Are you taking a Differential Equations class? If so you should have learned how to solve "second order linear equations with constant coefficients". Do you see that the "characteristic equations" are r^2= 1 for the first and s^2= a for the second? Do you know what to do with those?

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Julio said:
Find the general solution of the ODE:

$\dot{X_1}=X_1$

$\dot{X_2}=aX_2$

where $a$ is a constant.

Now fix it. Can you help me now?

Hint: What is the derivative of $$\displaystyle A e^{Bt}$$?

-Dan

So it's a single dot- a first derivative. That's even easier. Note that $X_1'= \frac{dX_1}{dt}= X_1$ can be written as $\frac{dX_1}{X_1}= dt$ and $X_2'= aX_2$ can be written as $\frac{dX_2}{X_2}= adt$.

To "solve" such a differential equation, to go from the derivative of a function to the function itself, integrate both sides! That's why topsquark asked "what is the derivative of $Ae^{Bt}$?".

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HallsofIvy said:
So it's a single dot- a first derivative. That's even easier. Note that $X_1'= \frac{dX_1}{dt}= X_1$ can be written as $\frac{dX_1}{X_1}= dt$ and $X_2'= aX_2$ can be written as $\frac{dX_2}{X_2}= adt$.

To "solve" such a differential equation, to go from the derivative of a function to the function itself, integrate both sides! That's why topsquark asked "what is the derivative of $Ae^{Bt}$?".

Thanks Hallsoflvy :)

$X(t)= \begin{pmatrix} X_1\exp(t)\\ X_2\exp(at) \end{pmatrix}$

is correct?

It's easy to check isn't it? If $X_1(t)= X_1e^t$ then $X_1(x)'= X_1e^r= X_1(x)$.
If $X_2(t)= X_2e^{at}$ then $X_2(x)'= X_2(ae^{at})= aX_2$.

Yes, those satisfy both equations. Well done!

(Personally, I would not use "X1" as the constant in a funtion I had already called X1(t)!)

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## 1. What are first order linear equations?

First order linear equations are mathematical equations that involve a first derivative, or rate of change, and are linear in nature. This means that the dependent variable (usually denoted as y) is directly proportional to the independent variable (usually denoted as x).

## 2. How are first order linear equations solved?

First order linear equations can be solved using a variety of methods, such as separation of variables, integrating factors, and substitution. The specific method used depends on the form of the equation and the given initial conditions.

## 3. What are solutions of the ODEs?

Solutions of the ODEs (Ordinary Differential Equations) are functions that satisfy the given equation and its initial conditions. They represent the behavior or evolution of a system over time.

## 4. Why are solutions of the ODEs important?

Solutions of the ODEs are important because they allow us to model and predict the behavior of various systems in fields such as physics, engineering, and biology. They also have numerous real-world applications, such as in population growth, chemical reactions, and electrical circuits.

## 5. Can first order linear equations have multiple solutions?

Yes, first order linear equations can have multiple solutions. This can occur when the equation is separable and has multiple solutions for the constant of integration, or when the equation has a general solution that includes multiple arbitrary constants.

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