An approximation of a differential equation

[KFC]

Homework Statement

Find an approximate oscillating soution of y'' = (y-x)^2 - exp(2*(y-x))

where y=y(x), y'' denotes second deriviate

Homework Equations

y'' = (y-x)^2 - exp(2*(y-x))

The Attempt at a Solution

I try to change a variable with p = y-x, so
dy/dx = dp/dx+1
y'' = p''

The original equation becomes p'' = p^2 - exp(2p)
when p^2 close to exp(2p) or p close to exp(p), we have p'' -> 0, i.e. p -> ax+b

I think the approximate solution is p = ax + b, but it turns out to be not. Does anyone give me some hint to find out the approximate oscillating solution? Thanks in advance

 

[mighty2000]

!

Hello,

Thank you for sharing your attempt at solving this problem. Your approach of changing variables to p = y-x is a good starting point. However, your assumption that p'' = 0 when p^2 is close to exp(2p) or p is close to exp(p) is not correct. In fact, when p is close to exp(p), p'' will be very large and not equal to 0.

To find an approximate oscillating solution, you can use the method of perturbation. This involves expanding the solution in a series and solving for each term. In this case, you can start by assuming a solution of the form y = a(x)cos(b(x)) where a(x) and b(x) are functions to be determined. Then, you can substitute this into the original equation and solve for a(x) and b(x) by equating coefficients of different powers of x. This will give you an approximate solution in terms of a(x) and b(x).

Alternatively, you can use numerical methods to solve the equation and find an approximate oscillating solution. This involves using a computer program or software to solve the equation numerically and obtain the solution.

I hope this helps and good luck with your solution!