Chaotic Junkyard

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)

 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 

Important Series in solving PDEs

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

 Question paper: https://drive.google.com/file/d/1q2L8zJSRsLfpaupx38m5tU25EueB1lwT/view?usp=sharing #DSE #PastPaper #M2math

Question paper:  https://drive.google.com/file/d/1JMmsQcj8Qjn7fs3i1BAYH3wPAakudgPO/view?usp=sharing Marking Scheme: https://drive.google.com/file/d/1JMmsQcj8Qjn7fs3i1BAYH3wPAakudgPO/view?usp=sharing #HKDSE #PastPaper