Which of the following differential equations is/are nonlinear ?

To determine which of the given differential equations are nonlinear, let us first define the criteria for a differential equation to be linear.

An ordinary differential equation of order n for the dependent variable $x(t)$ with respect to the independent variable $t$ is linear if it can be written in the form:
$$a_n(t)\frac{d^{n}x}{d{t}^n}+a_{n-1}(t)\frac{d^{n-1}x}{d{t}^{n-1}}+\cdots +a_1(t)\frac{dx}{dt}+a_0(t)x=g(t)$$
This definition implies two essential conditions for linearity:
1. The dependent variable $x$ and all its derivatives must be raised to the first power only.
2. There must be no products of the dependent variable $x$ and/or its derivatives (e.g., no terms like $x\frac{dx}{dt}$ or ${\left(\frac{dx}{dt}\right)}^2$).
3. The dependent variable $x$ and its derivatives must not be arguments of nonlinear functions (e.g., no terms like $e^x$, $\sin \left(x\right)$, or $e^{dx/dt}$).

Let us evaluate each of the given options against these criteria:

1. Option 1:
$$tx(t)+\frac{dx(t)}{dt}=t^2{e}^t$$
Here, both $x(t)$ and its first derivative $\frac{dx}{dt}$ are of first degree. The coefficients $t$ and $1$ depend only on the independent variable $t$. Therefore, this equation is linear.

2. Option 2:
$$\frac{1}{2}{e}^t+x(t)\frac{dx(t)}{dt}=0$$
This equation contains the term $x(t)\frac{dx}{dt}$, which is a product of the dependent variable $x$ and its derivative. Because of this product term, the equation is nonlinear.

3. Option 3:
$$x(t)\cos t-\frac{dx(t)}{dt}\sin t=1$$
Here, $x(t)$ and $\frac{dx}{dt}$ are both of first degree, and their coefficients $\cos t$ and $\sin t$ are functions only of the independent variable $t$. Therefore, this equation is linear.

4. Option 4:
$$x(t)+e^{\left(\frac{dx(t)}{dt}\right)}=1$$
This equation contains the term $e^{(dx/dt)}$, which is an exponential function of a derivative of the dependent variable. Since the derivative is inside a transcendental/nonlinear function, this equation is nonlinear.

Thus, the nonlinear differential equations are:
$$\frac{1}{2}{e}^t+x(t)\frac{dx(t)}{dt}=0,x(0)=0$$ and
$$x(t)+e^{\left(\frac{dx(t)}{dt}\right)}=1,x(0)=0$$