DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS

1. definition: f(y'''''''''... y'', y', y)=g(x), '''': is said to be the order of differential equations.

2. First order differential equations: y'+Py=Q, P and Q are the functions of x only

   • General solutions GS: 1.rearrange to the standard form (sometimes we need to use substitution) and use following methods
     • 2.seperating the variables
     • 2.integrating equations by exact equations founded by: multiplying functions by R=e^(∫Pdx)
       • --> d/dx(Ry)=R'y+Ry'=Py+y'. WTF: derived...?

3. Second order differential equations: ay''+by'+cy=f(x), where a b c are constants.

   • General solutions GS: CF+PI
     • CF: by auxiliary equation E.A.: y=Ae^αx +Be^βx; y=(A+Bx)e^αx; y=(Asinβx+Bcosβx)e^αx where sol=α±βi
     • PI: GUESS WORK: sub P.I. and if not works try to increase the index* of your substitution as the equaiton shouldn't be homogenous eg. y=Cu+D--> y=Cu^2+Du+G ???
       • Polynominal: y= An x^n + An-1 x^(n-1) +... +A1 x + Ao. eg. y=2u --> sub y=Cu+D
       • Natural log e: y= Ce^kx, known k
       • Trignometry function: y= Csinkx + Dcoskx, known k
   • 对于 ax^2 y''+bx y'+cy=f(x)...等形式, 找到substitution 转化成标准形式.

4. Linear equations complement

   • so firstly we set there is only one variable, (if equation contains x, y. you only consider x as variable, y as constant): if its index = 1, then it is linear.
   • properties
     • f(x1+x2)=f(x1)+f(x2)
     • answer and equation are correspondingly linear

5. Implicit Differentiation complements

   • d/dx(dz/dx)=(d''z/d''x)(dz/dx)
     • source from 'product rule' ~
   • d/dx(dz/dt)=(d''z/dt'')(dt/dx)
   • dy/dx=1/(dx/dy)