Solving a Nonlinear Differential Equation: A Step-by-Step Guide

[Semo727]

Hello!
I tried to prove, that ideal rope (see picture in attachment) has a shape of the function ch[x]. I finished with this equation

$$[h'(x)]^2-h(x)\cdot h''(x)+1=0$$

Yes, when you try function h[x]=ch[x], you get 0 on the left side, but I have no clue how to solve this equation (find the solution without knowing the solution). Even Mathematica has some problems, if I set boundary conditions. Could you please write how to solve this DE (providing it isn't too complicated, because I don't know much about solving nonlinear DE)

 

[BoTemp]

Guess and check is the only guaranteed way of finding a solution to any DE. You might be able to use a series expansion here (guess solution of form h(x) = sum(C_n * x^n) and plug in, but I'm not sure. That method is guaranteed to work for linear equations, non-linear it can get tricky.

 

[tacman]

Hello! Solving nonlinear differential equations can be a bit tricky, but with a step-by-step approach, it can be simplified. Let's break down the process of solving this specific equation.

Step 1: Rewrite the equation
The first step is to rewrite the equation in a standard form. In this case, we have:

[h'(x)]^2 - h(x)h''(x) + 1 = 0

We can rewrite this as:

[h'(x)]^2 = h(x)h''(x) - 1

Step 2: Use substitution
Now, let's make a substitution to simplify the equation. We can substitute u = h'(x). This will give us:

u^2 = h(x)u' - 1

Step 3: Solve for u
Next, we can solve for u using separation of variables. This means we can write the equation as:

u' = (u^2 + 1)/h(x)

We can then integrate both sides with respect to x:

∫u' dx = ∫(u^2 + 1)/h(x) dx

This will give us:

u = ∫(u^2 + 1)/h(x) dx

Step 4: Substitute back
Now, we can substitute back our original variable h(x) using our substitution u = h'(x). This will give us:

h'(x) = ∫(h'(x)^2 + 1)/h(x) dx

Step 5: Solve for h(x)
We can now solve for h(x) by integrating both sides once again:

∫h'(x) dx = ∫∫(h'(x)^2 + 1)/h(x) dx dx

This will give us:

h(x) = ∫∫(h'(x)^2 + 1)/h(x) dx dx

Step 6: Simplify
Finally, we can simplify the equation using the identity ch(x) = (e^x + e^-x)/2. This will give us:

h(x) = ∫∫(h'(x)^2 + 1)/h(x) dx dx = ∫∫(u^2 + 1)/(e^x + e^-x) dx dx

This is the general solution to the equation. From here, you can apply any boundary conditions or initial conditions to find