# Exploring integration possibilities for a higher-order differential equation

Problem,

$$U^{5}U^{(5)}+5U^{4}U^{\prime}U^{(4)}+5U^{4}U^{(2)}U^{(3)}+\frac{5}{2}U^{3}(U^{\prime})^{2}U^{(3)}+cU^{\prime}=0$$

$$U = U(x)$$,$$c$$ is a constant, and I want to know if the left-hand side of the given differential equation corresponds to the derivative of a function. I am not aiming to solve the differential equation; instead, I intend to simplify it through integration for ease of further analysis. In fact, this process is one step of the 'tanh method'.

My attempts,

It is easy to see that the antiderivatives of $$U^{5}U^{(5)}+5U^{4}U^{\prime}U^{(4)} + cU^{\prime}$$ is $$U^{5}U^{(4)} + cU$$. But $$5U^{4}U^{(2)}U^{(3)}+\frac{5}{2}U^{3}(U^{\prime})^{2}U^{(3)}$$ seems impossible to be the derivative of a function. So I tried again to check if the left-hand side is the second derivative of some function, but this idea was quickly ruled out. However, I want to know if there exists a function $$F$$, such that $$\frac{dF}{dx}=U^5U^{(5)}+5U^4U'U^{(4)}+5U^4U^{(2)}U^{(3)}+\frac52U^3(U')^2U^{(3)}+cU'$$

Here's a hint for analyzing the equation:

You should look into the following triangle, where $$U^{(n)}(x+k) = U^{(n-1)}(x+k+1)-U^{(n-1)}(x+k)$$:

$$U(x),U(x+1),U(x+2),U(x+3),U(x+4),U(x+5)$$ $$U'(x),U'(x+1),U'(x+2),U'(x+3),U'(x+4)$$ $$U^{(2)}(x),U^{(2)}(x+1),U^{(2)}(x+2),U^{(2)}(x+3)$$ $$U^{(3)}(x),U^{(3)}(x+1),U^{(3)}(x+2)$$ $$U^{(4)}(x),U^{(4)}(x+1)$$ $$U^{(5)}(x)$$

Some of the values in this triangle are constrained by the equation.

• Using this iteration, the formula seems to become more complex and can hardly use integration by parts I think? Commented Jan 20 at 2:47
• Using the triangle, integration is just "moving from line with U'(x), to a line with U(x)"
– tp1
Commented Jan 20 at 15:13