Good so far. So r = -3 and r = 1. Therefore two linearly independent solutions would be \(\displaystyle y = e^{-3t}\) and \(\displaystyle y = e^{t}\).karush said:$\tiny{3.1.1}$
find the general solution of the second order differential equation.
$$y''+2y'-3y=0$$
assume that $y = e^{rt}$ then,
$$r^2+2r-3=0\implies (r+3)(r-1)=0$$
new stuff... so far..
$$y=c_1e^{-3t}+c_2e^{t}$$topsquark said:Good so far. So r = -3 and r = 1. Therefore two linearly independent solutions would be \(\displaystyle y = e^{-3t}\) and \(\displaystyle y = e^{t}\).
One last little bit for you. We have two linearly independent solutions to a second order linear homogenous differential equation. How do you write down the general solution?
-Dan
Yes, that's it.karush said:$$y=c_1e^{-3t}+c_2e^{t}$$
guess that's it?
what is linear independent solutions?
A second order differential equation is a mathematical equation that involves a function and its second derivative. It is commonly used to model physical phenomena in science and engineering.
To find the general solution of a second order differential equation, you need to solve for the function that satisfies the equation. This can be done by using techniques such as separation of variables, variation of parameters, or the method of undetermined coefficients.
A general solution is a solution that satisfies the given differential equation for all possible values of the independent variable. On the other hand, a particular solution is a specific solution that satisfies the equation for a specific set of initial conditions.
The general solution of a second order differential equation can be used to model and predict the behavior of physical systems in science. It can also be used to analyze and understand the underlying mathematical principles of various phenomena.
Yes, the general solution of a second order differential equation can be applied to real-world problems in science and engineering. It can be used to solve problems related to motion, electrical circuits, and other physical systems.