Solving a Second Order Piecewise Quadratic Equation

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!

[Walter Roberson]

Is that x double-prime, quantities individually squared, quantities then individually multiplied by themselves?

Is that x double-prime, quantities individually squared, matrix-multiply by itself (inner product)?

Is that x double-prime, matrix-multiply by itself (inner product) to do the part, then again matrix-multiplied by itself?

That is, I am not clear as to why you are not using ?

[MarshallSc]

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.

Answer 1

[Walter Roberson]

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.