Diff Eq Exam 1 Review

## Integrating Factor

### Standard Form

$\frac{dy}{dt}+P(t)y=Q(t)$

### Solution

Integrating Factor 指：$e^{\int{P(t)}dt}$。我们在标准形式的两边同时乘这个式子，通过几步简单的运算，我们可以得到：

$(y{\cdot}e^{\int{P(t)}dt})'=Q(t){\cdot}e^{\int{P(t)}dt}$

$y{\cdot}e^{\int{P(t)}dt}=\int{Q(t){\cdot}e^{\int{P(t)}dt}}dt$

$y=\frac{\int{Q(t){\cdot}e^{\int{P(t)}dt}}dt}{e^{\int{P(t)}dt}}$

## Seperation of Variables

### Standard Form

$P(y)dy=Q(x)dx$

## Bernoulli Differential Equations

### Standard Form

$\frac{dy}{dt}+P(t)y=Q(t){\cdot}y^n$

### Solution

$\text{Let }v=y^{1-n}$

${\therefore}v'=(1-n)y^{-n}{\cdot}y'$

$y'=\frac{v'}{(1-n)y^{-n}}$

## Modeling

### Trick/Hint

$\frac{dy}{dt}=\text{Rate in}-\text{Rate Out}$

## Exact Equation

Exact Equation 是一种愚蠢奇妙的等式。

### Standard Form

$M_y=-N_x$

### Solution

$\int{M(x,y)}dx=\int{N(x,y)}dy$

$P(x,y)+C_y=Q(x,y)+C_x$

Example: If $P(x,y)=xln(y)+3y$ and $Q(x,y)=-xln(y)-24x$, then the solution would be:

$xln(y)+3y+24x=C$

## Population

### Standard Form

$P'(t)=rP(1-\frac{P}{K})$

Where P stands for the population, r stands for the growth rate and K stands for the carrying capacity, which means the max size of population that the environment can keep $P'(t)$ positive.

### Solution

$P=\frac{K}{Be^{-rt}+1}$

End - of - File