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 
    \(\int M(x,y)dx+ \int N(x,y)dy=c.\)
  7. Radioactive Decay
    \({dy \over dt}=-ky,\)
    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  .