# Questions tagged [nonlinear-system]

In mathematics, a nonlinear system of equations is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.

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### A question on the qualitative analysis of solution of a system of ODEs [closed]

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a non-zero smooth vector field satisfying $\text{div} f \ne 0.$ Which of the following are necessarily true for the ODE: $\dot{\mathbf{x}}=f(\mathbf{x})$? (a) ...
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### simultaneous non linear equations [closed]

$x^3y^3(x^3+y^3) = 905$ and $x^4y^4(x+y) = 810$. Find values for x and y. Divide the first equation by the second equation, and factor to give $(x+y)^2 = \frac{25}{6}$. If I had stopped at this point, ...
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### Solving Systems of Linear PDE's in R3 w/ Singular Matrix Coefficients

been curious whether first-order linear systems of PDE's of the form $A\vec{u}_{x}+B\vec{u}_{y}+C\vec{u}_{z}=\vec{0}$ can be solved in the case where all matrices "A, B and C" are singular, ...
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### Finding an analytical solution to a particular class of second order ODEs

Let $$\ddot{y}+f(x)y=0$$ Be a differential equation such that $\{y,f\}\subseteq\mathcal{C}^\infty$. We make the substitution $y=e^{u(x)}$ to get: $$\frac{d^2}{dx^2}e^{u(x)}+f(x)e^{u(x)}=0$$ Or (\...
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### how to show global asymptotic stability with $V(x)=f(x)^{T}Pf(x)$ as a lyapunov function.

consider the system $f(x)=\dot{x}$ with $f(0)=0$, $f(x)$ is continuously differentiable. $f(x)$ can be written as $f(x)=\int_{0}^{1}\frac{\partial f}{\partial x}(x\sigma)x\partial\sigma$ (The first ...
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### Example 1.1 Nonlinear Control Khalil

$f( x) =\begin{bmatrix} -x_{1} \ +x_{1} x_{2}\\ x_{2} \ -x_{1} x_{2} \end{bmatrix}$ is continuously differentiable on $R^2$. Hence, it is locally Lipschitz on $R^2$. It is not globally Lipschitz since ...
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