Solve 3rd Order DE: How Many Linearly Independent Solutions?

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In summary, there are three linearly independent solutions to a third order linear differential equation. This is because the set of all solutions to a linear, homogeneous, differential equation of order n forms a vector space of dimension n. The fundamental solutions, which satisfy certain conditions, are shown to be linearly independent and span the space.
  • #1
bigdirtycarl
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This is a quick one: Is there a way to tell how many linearly independent solutions there are to a differential equation? Like by its order? I have to solve a third order differential equation, and I am wondering how many possible linearly independent solutions there are. Three perhaps? Thanks!
 
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In general no, though if the equation and initial conditions meet some conditions the numbers of constants will equal the order.
 
  • #3
In order to talk about "linearly independent" solutions, you must be talking about linear differential equations.

The set of all solutions to a linear, homogeneous, differential equation, of order n, forms a vector space of dimension n so, yes, there are 3 "linearly independent" solutions to a third degree linear differential equation.

To prove this, consider the "fundamental" solutions, at some x= a. For an n degree equation, the fundamental solutions are solutions to the differential equation satisfying:
for [itex]y^(m)_n(a)= \delta_{mn}[/itex]. That is, the nth derivative of [itex]y_n[/itex] at x= a is 1 and all other derivatives are 0 there. (The "0" derivative is the value of the function.)

It is easy to show that those are "linearly independent" functions. Further if y(x) is any solution to the differential equation and \(\displaystyle Y_n\) is the nth derivative of y at x= a, then \(\displaystyle y(x)= \sum_{i= 0}^n Y_i y_i(x)\) so they also span the space.
 

What is a third order differential equation?

A third order differential equation is an equation that contains a third derivative of a function. It can be written in the form of y'''=f(x,y,y',y'').

What does it mean to solve a third order differential equation?

Solving a third order differential equation means finding the function that satisfies the equation. This function is called the solution, and it can be used to model real-world phenomena or make predictions.

How many linearly independent solutions can a third order differential equation have?

A third order differential equation can have up to three linearly independent solutions. This means that the three solutions are not multiples of each other and cannot be written as a combination of each other.

How do I know if my third order differential equation has three linearly independent solutions?

If the coefficients of the equation are constants (do not contain any variables), then the equation will have three linearly independent solutions. Additionally, if the roots of the characteristic equation (the equation obtained by setting the coefficient of the highest derivative to zero) are distinct, then the equation will also have three linearly independent solutions.

Why is it important to find the linearly independent solutions of a third order differential equation?

The linearly independent solutions of a third order differential equation provide a complete and unique solution. This means that any other solution can be written as a linear combination of these solutions. They also help in understanding the behavior and properties of the system described by the differential equation.

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