I was given a problem, it has been worded as follows: Use the substitution z = y

1. Can someone help me solve the following system of differential equations?
$\begin{array}{rcl}\frac{d{P}_0}{dt}& =& -C_1\lambda P_0{P}_1+\frac{C_1}{2}{P}_1^2\\ \frac{d{P}_1}{dt}& =& -C_2{P}_1+C_1\lambda P_0{P}_1-\frac{1}{2}(1+\lambda )C_1{P}_1^2+C_1{P}_1{P}_2\\ \frac{d{P}_2}{dt}& =& C_2{P}_1+\frac{C_1}{2}\lambda P_1^2-C_1{P}_1{P}_2\end{array}$
2. (Extending above) Suppose we have $K$ first order non-linear differential equations (similar to above), is there any simple method to solve them analytically?

The Laplace transform of $tu(t-2)=e^{-2s}\left[\frac{s+2s^2}{s^3}\right]$
True or False?

What is the Laplace transform of ${\displaystyle x\left(t\right)=e^{-t}\cos \left(\pi \right)t}$?

Find the general solution to the following second order differential equations.
${\displaystyle y{}^{″}-6^{\prime }+25y=0}$

Find the differential equations of the following. Show complete solution. ${\displaystyle \frac{dy}{dx}+y=e^{2x}}$

I have a first order PDE:
$x{u}_x+(x+y)u_y=1$
With the initial condition:
I have calculated result in Mathematica: $u(x,y)={\displaystyle \frac{y}{x}}$, but I am trying to solve the equation myself, but I had no luck so far. I tried with method of characteristics, but I could not get the correct results. I would appreciate any help or maybe even whole procedure.

Most systems can be modeled using first order system just like a temperature control system of an incubator.
Question 1: How can we know that a certain system can be modeled using first order differential equation?
${\displaystyle a_1\frac{dy}{dt}+a_{0}y=F\left(t\right)\left(1\right)}$
A first order differential equation in Equation (1) may be written as:
${\displaystyle \tau \frac{dy}{dt}+y=kF\left(t\right)\left(2\right)}$
Question 2: If a step input
${\displaystyle F\left(t\right)=\{\begin{array}{cc}0& \text{if}t<0\\ A& \text{if}t>0\end{array}\left(3\right)}$
of the system has an output response of a
${\displaystyle y\left(t\right)=kA(1-e^{\frac{-t}{\tau }}\left(4\right)}$
then the system can be modeled as first differential equation where
${\displaystyle y_{max}=kA\left(5\right)}$
Is this correct?
Given a step input and the graph of the step response of the system. Am i correct if I say that if the output of the system behaves similar to equation (4), then I can model the system using first differential equation?