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

[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.

y_h(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..

 

[Unco]

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).

 

[leyyee]

hey there.
thanks a lot for the reply..

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

 

[leyyee]

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) = u_h(x) + u_p(x) = C₁e^2x+C₂e^-2x+8xe^2x-2e^2x

how should I convert it to y(x) = y_h(x) + y_p(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

 

[Unco]

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) = u_h(x) + u_p(x) = C₁e^2x+C₂e^-2x+8xe^2x-2e^2x

Well done!

how should I convert it to y(x) = y_h(x) + y_p(x)

No need; you've done the variation of parameters work for u.

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

Exactly!

 

[leyyee]

Wow.. thanks a lot.. 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