Free Fall with Air Resistance

The 2nd Order ODE is non-Linear in the form of:

The intuitive solution is to convert it as a first-order ODE (still non-linear) with
 

It reduces into:

By Separable:


It could be solved by trigonometric substitution (using sine preferred) and result in:


Remaking the subject:



The integration could be done by the substitution:


This makes the integral into:


The integration could be solved easily by considering:


Then the solution follows:


Finalized:

 


#ODE #AirResistance






留言

張貼留言

這個網誌中的熱門文章

Evaporation (in Conical flask)

圖片
 By definition of the evaporation, the rate of volume change shall be directly proportional to that of the surface area exposed: In this case assume that the water is evaporating in a conical flask. The evaporation is done by the upper surface exposed to the air (the surface will always be a circle).  The volume of water at any time t would be enclosed in the shape of a frustum as shown. The volume is hence: Subject to the needs of the differential equation, we rearrange it in the form of Area as a function of Volume, By substitution, In which 
閱讀完整內容

Canonical Forms and General Solutions of 2nd PDE

圖片
Consider the second-order equation in which the derivatives of second-order all occur linearly, with coefficients only depending on the independent variables: Step 1: Solving the characteristic equation is: Step 2: Solve the 1 st order differential equation. Step 3: Define μ(x, y) and η(x, y) as the two “constant” function as the result of the Step 2 Step 4: u = u(μ(x, y), η(x, y)) Step 5: Conduct partial differentiation and substitute back the result into the original PDE, resulting in a Canonical Form. Step 6: Integrate (partial) the simple transformed Canonical Form to arrive General Solution (Note that if all coefficients are Constants, final answer: u =  μ(x, y) +  η(x, y), as the general solution)
閱讀完整內容