# 5th order DE with g(x)=32exp^(2x)

• leyyee
In summary, the student attempted to solve the homework equation but is not sure if they are in the right path. They found the wronskian matrix and determined the determinant, but are not sure if they are in the correct path to find the general solutions.
leyyee

## Homework Statement

Use variations of parameters to find the general solutions of the following differential equations.

y'''''-4y'''=32exp^(2x)

## Homework Equations

no relevant equations.

## The Attempt at a Solution

hey there, I tried solving this question. I got the homogeneous equation.

yh(x)=C1+C2x+C3X2+C4exp^(2x)+C5exp^(-2x)

but after this step.. I am stuck..

Because I am not quite sure whether I am in the correct path to look for the general solutions. After I have this equation, I did the wronskian matrix, I found the determinant of the 5X5 matrix is 512. Am I correct? Please correct me if I am not in the journey to my answer.

Besides that , If I were to use the variations of parameters. The matrix would be 5X5 dimensions. Is it correct?

Thanksedited due to duplication on the template question.. sorry..

You should get into the habit of reducing such a DE into something more familar. Let $$u=y^{(3)}$$, so the DE becomes $$u'' - 4u = 32e^{2x}$$.

Proceed as usual (involves a 2x2 matrix) to solve for u(x). Then you can obtain y(x).

hey there.
thanks a lot for the reply..

i will try out your attempt.. then i will post if i need anymore help..
thanks

hey there, i have tried out your attempt.. but I only can find the general solutions

the wronskians i found was -4, am I in the right path ?

u(x) = uh(x) + up(x) = C1e2x+C2e-2x+8xe2x-2e2x

how should I convert it to y(x) = yh(x) + yp(x)

If i use the equation u gave me u = y(3). Am I suppose to integrate u(x) by 3 times?any clue for me?

thanks

Last edited:
leyyee said:
hey there, i have tried out your attempt.. but I only can find the general solutions

the wronskians i found was -4, am I in the right path ?

u(x) = uh(x) + up(x) = C1e2x+C2e-2x+8xe2x-2e2x
Well done!

leyyee said:
how should I convert it to y(x) = yh(x) + yp(x)
No need; you've done the variation of parameters work for u.

leyyee said:
If i use the equation u gave me u = y(3). Am I suppose to integrate u(x) by 3 times?
Exactly!

Wow.. thanks alot.. I think I will post the proper solutions after I finished the steps ok ?

and I will let you go through my answer to see whether I did any mistakes.. Thanks

## 1. What is a 5th order differential equation?

A 5th order differential equation is a mathematical equation that involves the 5th derivative of an unknown function. It is a type of differential equation that is used to model physical systems and phenomena in various fields such as physics, engineering, and biology.

## 2. What is the role of g(x) in a 5th order differential equation?

In a 5th order differential equation, g(x) represents a forcing function or an external input that affects the behavior of the system. It can be a constant or a function of the independent variable x.

## 3. How is a 5th order differential equation solved?

The process of solving a 5th order differential equation involves finding a solution for the unknown function by integrating the equation multiple times. The initial conditions and any given boundary conditions are also used to determine the final solution.

## 4. What is the significance of e^(2x) in the given 5th order differential equation?

e^(2x) is a term that is commonly used in differential equations to represent exponential growth or decay. In this particular equation, it is used as part of the forcing function g(x), indicating that the system is influenced by an exponential growth or decay process.

## 5. How are 5th order differential equations used in real-world applications?

5th order differential equations are used to model complex systems and phenomena in various fields such as physics, engineering, and biology. They are also used in mathematical modeling and simulation to predict and understand the behavior of these systems under different conditions.

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