Higher Order Linear Homogenous Differential Equation

In summary, the conversation discusses a boundary value problem and the process of finding a general solution. The person seeking help asks for confirmation that their solution is correct and receives advice to plug it back into the original equation to verify. They also mention applying initial and boundary conditions and express gratitude for the help and the suggested method for double-checking solutions.
  • #1
Mark Rice
37
0

Homework Statement


Hi, basically I have a boundary value problem and just want to check that my general solution is correct.

x'''' + 16x = 0

Homework Equations

The Attempt at a Solution


I'm pretty sure you make a characteristic equation which would be m4 + 16 = 0.
Solving this I get m to be √2 +- √2 i and -√2 +- √2 i. I therefore get my general solution to be:

Ae(√2t)cos(√2t) + Be(-√2t)cos(√2t) + Ce(-√2t)sin(√2t) + De(√2t)sin(√2t)

Is this correct or am I on totally the wrong track? I just want to make sure this is correct before applying the initial and boundary coniditions. Thanks.
 
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  • #2
Mark Rice said:

Homework Statement


Hi, basically I have a boundary value problem and just want to check that my general solution is correct.

x'''' + 16x = 0

Homework Equations

The Attempt at a Solution


I'm pretty sure you make a characteristic equation which would be m4 + 16 = 0.
Solving this I get m to be √2 +- √2 i and -√2 +- √2 i. I therefore get my general solution to be:

Ae(√2t)cos(√2t) + Be(-√2t)cos(√2t) + Ce(-√2t)sin(√2t) + De(√2t)sin(√2t)

Is this correct or am I on totally the wrong track? I just want to make sure this is correct before applying the initial and boundary coniditions. Thanks.
Take your general solution and plug it back into the original ODE. If you're on the right track, you'll know it when you get zero on both sides of the equation.
 
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  • #3
SteamKing said:
Take your general solution and plug it back into the original ODE. If you're on the right track, you'll know it when you get zero on both sides of the equation.
What SteamKing suggests is something you should always do when you're working with diff. equations.
 
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  • #4
Right cool thanks guys!
 
  • #5
Cool, I plugged it in and got 0 (99% sure I did this correctly, there was a lot of terms haha!), so that means it is the correct and I can move on to applying the boundary functions? Thanks for all the help, will definitely use that plugging in method to double check my answers in future :)
 
  • #6
Yep, plug ahead.
 
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Related to Higher Order Linear Homogenous Differential Equation

1. What is a higher order linear homogenous differential equation?

A higher order linear homogenous differential equation is a type of differential equation that involves the derivatives of a function, where the highest order of derivative is greater than 1 and the coefficients of the derivatives are constants. It is called "homogenous" because all terms in the equation only contain the dependent variable and its derivatives, without any independent variable.

2. What is the general form of a higher order linear homogenous differential equation?

The general form of a higher order linear homogenous differential equation is:
an(x)y(n) + an-1(x)y(n-1) + ... + a1(x)y' + a0(x)y = 0
where an(x), an-1(x), ..., a1(x), a0(x) are functions of the independent variable x, and y(n) represents the nth derivative of the function y with respect to x.

3. How do you solve a higher order linear homogenous differential equation?

To solve a higher order linear homogenous differential equation, you can use the method of undetermined coefficients or the method of variation of parameters. In the method of undetermined coefficients, you assume a solution of the form y = erx, where r is a constant, and solve for r using the characteristic equation. In the method of variation of parameters, you assume a solution of the form y = u1(x)y1(x) + u2(x)y2(x) + ... + un(x)yn(x), where y1(x), y2(x), ..., yn(x) are known solutions of the homogenous equation and u1(x), u2(x), ..., un(x) are unknown functions to be determined.

4. How do higher order linear homogenous differential equations relate to real-world problems?

Higher order linear homogenous differential equations are commonly used to model real-world problems in physics, engineering, and economics. For example, the motion of a mass on a spring can be described by a second order linear homogenous differential equation, where the displacement of the mass is the dependent variable and the force of the spring is the independent variable. In economics, higher order linear homogenous differential equations can be used to model the relationship between supply and demand in a market.

5. What are some applications of higher order linear homogenous differential equations in science and mathematics?

Higher order linear homogenous differential equations have a wide range of applications in science and mathematics. They are used to model physical systems such as oscillating systems, electrical circuits, and fluid dynamics. In mathematics, they are used to study the behavior of functions and find solutions to various problems. They are also used in numerical methods for solving differential equations, which are essential in many fields of science and engineering.

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