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Julio1
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Find the general solution of the ODE:
$\check{X_1}=X_1$
$\check{X_2}=aX_2$
where $a$ is a constant.
$\check{X_1}=X_1$
$\check{X_2}=aX_2$
where $a$ is a constant.
Julio said:Find the general solution of the ODE:
$\dot{X_1}=X_1$
$\dot{X_2}=aX_2$
where $a$ is a constant.
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 $Ae^{Bt}$?".
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).
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.
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.
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.
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.