Ordinary Differential Equations and its applications

Ordinary Differential Equations and its applications

1. Classifications

   • Type
     • ODE of single independent variable
     • PDE of one or more dependent variable
   • Order
     • First
     • Second
     • n-th
   • Linearity
     • LINEAR if F is a linear functionof the dependendent variables as well as its derivatives
   • Homogeneity
     • F =0, then homogeneous
     • F =/= 0, then not homogeneous

2. Solutions to ODEs

   • Particular
     • A solution free of arbitrary parameters c
   • Singular
     • solution that cannot be obtained by specifying values of parameters in a family of solutions
   • General/Complete
     • EVERY solution of the DE

3. Picard's Theorem

   • STEP 1
     • Show f(x,y) is continuous
     • STEP 2
       • Show df/dy is also continuous
       • STEP 3
         • Shown that theorem is satisfied, therefore it is a unique solution
   • STEP 1

4. Sketching direction fields

   • STEP 2
     • Find constant/equilibrium solutions y=c, if any
     • STEP 3
       • Show it's concavity for each region
         • Compute y' <0 , y' = 0 , y' > 0
       • STEP 4
         • Draw the graph