Understanding the General Solution of a Differential Equation

[oneamp]

Hello - I asked a similar question before, but it was not resolved for me, and the person who answered was rude, so I did not continue the conversation.

I read this here: http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx

"If y_1(t) and y_2(t)are two solutions to a linear, homogeneous differential equation then so is
y(t) = c_1 y_1(t) + c_2 y_2(t), and it states that this is the general solution.

I don't understand this: if y_1(t) and y_2(t) are solutions, then we should be done, right? We have our solutions. Why are we interested in making another solution? And, why is the sum of the solutions with multipliers the "general solution", and the other two solutions, y_1(t) and y_2(t), not general?

Thank you

 

[pasmith]

Hello - I asked a similar question before, but it was not resolved for me, and the person who answered was rude, so I did not continue the conversation.

I read this here: http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx

"If y_1(t) and y_2(t)are two solutions to a linear, homogeneous differential equation then so is
y(t) = c_1 y_1(t) + c_2 y_2(t), and it states that this is the general solution.

I don't understand this: if y_1(t) and y_2(t) are solutions, then we should be done, right? We have our solutions. Why are we interested in making another solution? And, why is the sum of the solutions with multipliers the "general solution", and the other two solutions, y_1(t) and y_2(t), not general?

Thank you

It's not enough for a solution to satisfy the ODE; it must also satisfy the initial conditions, and for a second-order ODE there are two such conditions.

Example:

The functions $y_1(x) = \cos x$ and $y_2(x) = \sin x$ are solutions of
$$y'' = -y.$$

Neither $y_1$ nor $y_2$ satisfy the intitial conditions $y(0) = y'(0) = 1$, but the linear combination $y(x) = \cos x + \sin x = y_1(x) + y_2(x)$ does, and in general $a\cos x + b\sin x$ is the solution of $y'' = -y$ which satisfies the initial conditions $y(0) = a$ and $y'(0) = b$.

 

[oneamp]

Thank you

 

[HallsofIvy]

The basic theorem here is that "the set of all solutions to a linear homogeneous differential equation of order n form an n dimensional vector space".

In particular, that means there exist a "basis" for the vector space (solution set) consisting of n functions such that every solution can be written as a linear combination of those solutions.

In the case of a "linear homogeneous second order equation", there must exist two independent solutions, $y_1(x)$ and $y_2(x)$ such that any solution, y(x), can be written $y(x)= Ay_1(x)+ By_2(x)$ for appropriate constants A and B.

We are not "done" when we find $y_1$ and $y_2$ because there exist an infinite number of solutions.

 

[blue_raver22]

for your help."

I can help clarify the concept of the general solution of a differential equation for you. The general solution is a solution that encompasses all possible solutions to a given differential equation. In other words, it is a solution that can be used to generate any specific solution to the equation by simply choosing appropriate values for the constants c1 and c2.

In the context of linear, homogeneous differential equations, this general solution is given by the equation y(t) = c1y1(t) + c2y2(t), where y1(t) and y2(t) are two solutions to the equation. This is because, for linear equations, the sum of any two solutions is also a solution. Therefore, by choosing different values for c1 and c2, we can obtain different specific solutions.

You may be wondering why we need a general solution if we already have specific solutions in the form of y1(t) and y2(t). The reason is that in many cases, it is difficult or even impossible to find specific solutions to a differential equation. In such cases, the general solution provides a way to express all possible solutions without explicitly finding them. It is a more general and flexible approach.

In summary, the general solution is a powerful tool in solving differential equations as it allows us to express all possible solutions in a single equation. I hope this explanation helps clarify your understanding. If you have any further questions, please feel free to ask.