Solving a Second Order Piecewise Quadratic Equation

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How can I solve a second order quadraic nonlinear equation for each components of two matrices. For example, having:
a=rand(10,10); b=rand(10,10);
For the equation:
I'm looking for the analytical solution. I tried to write a code but I don't know what solver to use, ODE45 or dsolve.
Also, if I want to solve the gradient of this equation (getting the antiderivative of the equation to make it first order equation) which will be solmething like this:
How can I solve them? I'd appreciate it if someone can help me!
MarshallSc on 3 Dec 2021
Well, yes, since they are the same, it will be in the form of . I wrote it in this way because sometimes, the two xs are not the same (not in this case), and I was going to ask a follow up question to see how I can solve a set of these equations if the two xs are not the same (x1, x2) and we have another equation with different a, b (say c,d), so it will be a system of equations.

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Accepted Answer

Walter Roberson
Walter Roberson on 3 Dec 2021
syms a b x(t)
x__prime = diff(x)
x__prime(t) = 
x__prime__prime = diff(x__prime)
x__prime__prime(t) = 
eqn = a/(x__prime__prime*x__prime__prime) == b
eqn(t) = 
sol = dsolve(eqn)
sol = 
syms x_2(t)
x_2__prime = diff(x_2)
x_2__prime(t) = 
x_2__prime__prime = diff(x_2__prime)
x_2__prime__prime(t) = 
eqn_2 = a/(x__prime__prime * x_2__prime__prime) == b
eqn_2(t) = 
sol_2 = dsolve(eqn_2)
sol_2 = 
I think for sol_2 there is a fundamental problem that you are asking to solve one equation with respect to two functions.

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