Calculus 2 Equations: Expert Solutions and More

Dirac delta in ODE (inhomogenous), where is the discontinuity?
Trying to understand where the discontinuity comes from in this equation coming from an ODE course (link at the bottom):
${\displaystyle \ddot{x}+3\dot{x}+2x=\delta \left(t\right)}$,
Initial conditions: ${\displaystyle \dot{x}\left(0\right)=x\left(0\right)=0}$. Assume impulse occurs at ${\displaystyle t=0^{+}}$
The solution is: ${\displaystyle x\left(t\right)=e^{-t}-e^{-2t}=e^{-t}(1-e^{-t})}$
And the derivative of the solution evaluated at IC:
${\displaystyle \dot{x}(t=0)=-e^{-t}+2e^{-2t}=-1+2=1}$
The course mentions that the x(t) is continuous at ${\displaystyle t=0}$, but the derivative x˙(t) is not because ${\displaystyle \dot{x}\left(0^{-}\right)=0}$ whilst ${\displaystyle \dot{x}\left(0^{+}\right)=1}$. How do I know that ${\displaystyle \dot{x}\left(0^{-}\right)=0}$? Does one need to read into when impulse occurs to arrive at the conclusion or can it be shown mathematically using the Dirac's delta definition?

Is there a way, using the definition,
${\displaystyle \zeta \left(s\right)=\sum _{i=1}^{\mathrm{\infty }}{i}^{-s}}$
to find this?

How do I prove that ${\displaystyle \left(v_n\right)}$, defined by
${\displaystyle v_n=\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots +\frac{1}{\sqrt{n^2+n}},}$
is convergent?

Find the number of unique solutions of the following nonlinear ODE
${\displaystyle y^{\prime }=\frac{10}{3}x{y}^{\frac{25}{,}}y\left(0\right)=-1}$

Find the general solution
${\displaystyle x\frac{dy}{dx}-y=\frac{1}{x^2}}$

Does Lyapunov function need to be defined at zero?
Consider the function ${\displaystyle V(x,y)=x-y-y\ln \left(\frac{x}{y}\right)}$, where ${\displaystyle x,y>0}$ and y is fixed. V is constructed to study the stability of equilibrium point ${\displaystyle x=y}$ for the system
${\displaystyle \frac{dx\left(t\right)}{dt}=y-x\left(t\right).}$

How do you go about solving the differential equation of the following form?
${\displaystyle \frac{dy}{dt}+y\left(t\right)=k}$, where k is a constant.
Would the method of integrating factors typically used for the following forms work:
${\displaystyle \frac{dy}{dt}+a\left(t\right)y=b\left(t\right)}$

Differentiation question - (an application of similarity solutions)
${\displaystyle \psi =x^{\frac{1}{3}}f\left(\eta \right)}$
where,
${\displaystyle \eta =\frac{y}{x^{\frac{2}{3}}}}$

What is exponential growth in environmental science?

Radius of convergence of
$\underset{n=2}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}\frac{n^{2n}}{4^n(2n+1)!}(3-<!--\; -\; -->2x{)}^n$

How do you solve the differential ${\displaystyle y^{\prime }x\ln \left(x\right)=y}$?

How do you solve for xy'-y=3xy given y(1)=0?

How do you solve separable first-order differential equations?

How do you solve the differential equation ${\displaystyle y^{\prime }=e^{-y}(2x-4)}$, where y(5)=0?

Use the definition of stability of equilibrium to prove (1,0) is unstable.
Given the system
${\displaystyle x^{\prime }=x(1-r)+y(x-r)}$
${\displaystyle y^{\prime }=y(1-r)+x(r-x)}$
where ${\displaystyle r^2=x^2+y^2}$. I don't know how to use the definition of stability of equilibrium to show that the equilibrium (1,0) is unstable.

The solution to the initial value problem
${\displaystyle \frac{dy}{dx}=(x-2){(y-3)}^2,y\left(0\right)=0}$,
will always be less than 3; that is, ${\displaystyle y\left(x\right)<3}$ for ${\displaystyle x\ge 0}$.

How do you solve x''(t)+x3=0?

How do you solve this series question?
${\displaystyle \sum _1^{\mathrm{\infty }}\frac{\cos \left(n\pi \right)}{\ln \left(6n\right)}}$

How do you solve for the equation ${\displaystyle \frac{dy}{dx}=\frac{3x^2}{e^{2}y}}$ that satisfies the initial condition ${\displaystyle f\left(0\right)=\frac{1}{2}}$?