Integration by parts of 4th order DE

[shuttleman11]

having difficulty integrating the following equation by parts to determine if its symmetric:

d4 u / dx4 + K d2 u / dx2 + 6 = 0 0< x < 1

Can someone help with this?

 

[rock.freak667]

If K is a constant then you have a 4th order DE with constant coefficients. So that all of your answers will be in the form y=e^rx.

d⁴u/dx⁴ + K d²u/dx² + 6=0

 

[shuttleman11]

But I'm lost with assigning u, v, du, dv to this equation? Also can I split it into three idfferent integrals added together or must I assign the values and differentiate the hole equation by parts? Not for sure if I'm being clear?

 

[rock.freak667]

$$\frac{d^4u}{dx^4} + K \frac{d^2u}{dx^2} + 6=0$$

Now you can just integrate everything with respect to x to get

$$\int \frac{d^4u}{dx^4}dx + \int K \frac{d^2u}{dx^2}dx + \int 6dx= \int 0dx$$



Now $\int \frac{dy}{dx}dx=y+c$ where c is a constant. Now just use this idea to work out your problem.

 

[HallsofIvy]

But I'm lost with assigning u, v, du, dv to this equation? Also can I split it into three idfferent integrals added together or must I assign the values and differentiate the hole equation by parts? Not for sure if I'm being clear?

This is NOT a first order equation: you can't just integrate both sides. rockfreak667 told you to do it as an equation with constant coefficients. What is its characteristic equation?

$$\frac{d^4u}{dx^4} + K \frac{d^2u}{dx^2} + 6=0$$

Now you can just integrate everything with respect to x to get

$$\int \frac{d^4u}{dx^4}dx + \int K \frac{d^2u}{dx^2}dx + \int 6dx= \int 0dx$$


Now $\int \frac{dy}{dx}dx=y+c$ where c is a constant. Now just use this idea to work out your problem.

And what is $\int d^2y/dx^2 dx$?