Solve System of Differential Equations with Expert Help

A bacterial population B is known to have a rate of growth proportional to B itself. If between noon and 2pm the population triples, at what time no controls being exerted, should B becomes 100 times what it was at noon? what wiil be the population at 6pm?

${\displaystyle {\int }_{\partial \mathrm{\Omega }}\frac{\partial u}{\left[\partial N\right\}d\sigma \text{if}\mathrm{\Delta }u=0\text{in}\mathrm{\Omega }}}$?
Is it always possible to solve an equation ${\displaystyle u{}^{″}=0}$ in [0,1] if ${\displaystyle u^{\prime }\left(0\right)\text{and}{u}^{\prime }\left(1\right)}$ are given?(What is connection to first question?) Is the solution - unique?

Solve for G.S./P.S. for the following differential equations using the solutions by a change in variables as suggested by the equations.
${\displaystyle (t^2{e}^t-4x^2)dt+8txdx=0}$

I need help
Solve the following first order differential equation:
${\displaystyle 3t{x}^{\prime }-2x=\frac{t^3}{x^2}}$

Dont

The Runge Kutta method is known as:
a. An analytical method of solving first order differential equations.
b. An accurate method of solving first order differential equations.
c. A numberical method of solving first order differential equations.
d. A complex method of solving first order differential equations.

Need your help!
1. Identify the substitution method used to solve the given differential equations. Support your answer with the appropriate calculations.
2. Solve for the solution to the given differential equation that satisfies the given condition
${\displaystyle y^{\prime }=2{(3x+y)}^2-1;y\left(0\right)=1}$

Determine if the equation is linear or non-linear and indicate the order of the differential equation and would you tell me why please
${\displaystyle (2x^3+y)dx-xdy=0}$
${\displaystyle ydx-(x+6y^2)dy=0}$
${\displaystyle (x+y)dx+\tan \left(x\right)dy=0}$
${\displaystyle 6xydx+(4y+9x^2)dy=0}$

Classify the equation as separable, linear, both, or neither.
${\displaystyle y\frac{dy}{dx}=2x+y}$

Is the following is linear equation?
${\displaystyle {\left(\frac{dy}{dx}\right)}^2+\cos \left(x\right)y=5}$

How to solve ${\displaystyle a\ddot{u}+b{\left(\dot{u}\right)}^2+\dot{u}+\dot{u}c{e}^u+e^u-e^{2u}+1=0}$

How to solve ${\displaystyle y^{\prime }+y=e^x{y}^{\frac{2}{3}}}$?

For the sulotion of the OED: ${\displaystyle \frac{dy}{dx}=1+x+y^2+xy}$.

${\displaystyle f\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^{n}u(t-n)}$
show that ${\displaystyle L\left\{f\left(t\right)\right\}=\frac{1}{s(1+e^{-s})}}$

Solution to a differential equation
${\displaystyle x=-ytextandy=x}$

Find the values of a and b
Given that the curve ${\displaystyle y=a{x}^2+\frac{b}{x}}$ has a gradient of -5 at the point (2,-2), find the value of a and b.

Find the general solution of
${\displaystyle y^{IV}+2y+y=\cos x}$.

Solve: ${\displaystyle \frac{dp}{dt}=t^{2}p-p+t^2-1}$
I got ${\displaystyle \int \frac{1}{p+1}dp=\int (t^2-1)dt}$
After take the integral I got
${\displaystyle \ln |p+1|+c=\frac{1}{3}{t}^2-t+c}$
after this step I stacked. How can I simplify like ${\displaystyle p=}$ something?

Solving a differential equation by power series.
I want to find the power series solutions about the origin of two linearly independent solutions of
${\displaystyle w-zw=0}$.
Also, how do I show that these solutions are analytic?

Solve ${\displaystyle \frac{d^2\theta }{{dt}^2}+g\sin \theta =0}$