Ex 1. Ordinary Differential Equations

Calculus 2



Weiqi Zhou

Ordinary Differential Equations

  • Differential equations are equations that contain derivatives
    e.g.,

  • The order of the equation is the highest order of the derivate
    e.g., 1st order: 2nd order：

General and Particular Solutions

  • A general solution is an infinite family of solutions that satisfy the equation

  • A particular solution is a solution that satisfies a particular initial value condition:

Example

  • Population growth：；
    Solution：

  • Growth in pairs：；
    Solution：；

  • The logistic curve：；
    Solution：

It Describes Time Dependent Systems

Separable ODE

Solution

  • Rewrite it as

  • Integrate at both sides to get

  • If are anti-derivatives of and respectively

  • Then is a general solution，where is a constant

Example

  • Find the solution to

  • Rewrite as ，integrate to get

  • ，the minus sign can be absorbed into the constant

  • The general solution is ，plug in the initial value to get the particular solution

Exercise

Find the solution to

Homogeneous ODE

Homogeneity

  • If there exists so that holds for any ，then is called an order homogeneous function

  • Example：

  • is homogeneous

Solution

  • Let ，then is a function of ，

  • Plug it into the equaiton to get

  • Separating variables leads to

  • It is a separable ODE, solve and plug back in afterwards.

Example

  • Find the solution to

  • Rewrite it as ，let ，then

  • Separate variables: , and integrate:

  • ，hence the general solution is ，the particular solution is

Exercises

Find solutions to following equations

High Oder ODE

：integrate times

Exercises

  • Find the solution to satisfying

  • Find the solution to ()

  • Find the particular solution to that passes and is tangent to at this point

High Oder ODE

：convert to where

Example

  • Find the solution to

  • Let , then

  • Separate variables to get ，consequently

High Oder ODE

：Take

then ，hence

Example

  • Find the solution to

  • Let , then

  • Separate variables to get ，i.e.,

  • Hence

Exercises

  • Find particular solutions to following equaitons



Summary

  • Concepts of ODE

  • Separable ODE

  • Homogeneous ODE

  • High Order ODE