# Differential equations - First order ordinary differential equations

1. Linear Equation
${dy \over dx}+p(x)y=q(x).$
The general solution is
$y={{\int u(x) q(x)dx+c} \over u(x) },$
Where
$u(x)=exp(\int p(x)dx).$
2. Separable Equations
${dy \over dx}=f(x,y)=g(x)h(y)$
The general solution is given by
$\int { dy\over h(y) }=g(x)dx+c.$
or
$H(y)=G(x)+C$
3. Homogeneous Equations
The differential equation ${dy \over dx}=f(x,y)$ is homogeneous, if the function f(x,y) ishomogeneous, that is
$x{dz \over dx }+z=f(1,z)$
4. Bernoulli Equation
${dy \over dx}+p(x)y=q(x)y^n .$
The substitution $z= y^{1-n}$ leads to the linear equation
${dz\over dx}+(1-n)p(x)z=(1-n)q(x).$
5. Riccati Equation
${dy \over dx}=p(x)+q(x)y+r(x)y^2$
If a particular solution Y is know, then the general solution can be obtained with the help of substitution
$z={1 \over y-y_1},$which leads to the first order linear equation
${dz \over dx }=-[q(x)+2y_1r(x)]z-r(x).$
6. Exact and Nonexact Equations
The equation
M(x,y)dx+N(x,y)dy=0
is called exact if
${\partial M\over \partial y}={\partial N\over \partial x},$
and nonexact otherwise ,
The general solution is
$\int M(x,y)dx+ \int N(x,y)dy=c.$
${dy \over dt}=-ky,$
where y(t) is the amount of radioactive element at time t, k is the rate of decay.
The solution is
$y(t)=y_0e^{-kt},$Where $y_0=y(0)$ is the initial amount.
8. Newton's Law of Cooling
${dT \over dt}=-k(T-S),$
where $T(t)$ is the temperature of an object at time t, S is the temperature of the surrounding environment , k is a positive constant.
The solution is
$T(t)=S+(T_0-S)e^{-kt},$
where $T_0=T(0)$ is the initial temperature of the object at time t=0  .
9. Population Dynamics (Logistic Model )
${dP \over dt}=kP \Big( 1-{P\over M} \Big ),$
where $P(t)$ is population at time t,k is positive constant, M is limiting size for the population.
The solution of the differentiaal equation is
$P(t)={MP_0 \over P_0+(M-P_0)e^{-kt}},$ where $P_0=P(0)$ is the initiaal population at time t=0  .