Completed notes of the course Mathematical Methods

Page 1

1 NOTES OF MATHEMATICAL METHODS Prof. F. Gazzola – A.A. 2020/21 REVIEW OF ODEs ODEs ODEs are Ordinary Differential Equations in the form: 𝑦!=𝑓(𝑦) ,𝑓:𝐼∈ℝ→ℝ Linear first order ODEs The linear first order ODEs are differential equation can be solved in different ways depending on their particular form, but in general they have the following form: 𝑦!=𝛼𝑦+𝛽 Malthus equation The Malthus equation is the simplest linear first order ODE and its solution is often used during the resolution of more complicated ODEs: 𝐶𝑎𝑢𝑐ℎ𝑦 𝑝𝑟𝑜𝑏𝑙𝑒𝑚: <𝑦!=𝜀𝑦𝑦(0)=𝑦" → 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝑦(𝑡)=𝑦"𝑒#$ ´ Since 𝐹(𝑦)=𝜀𝑦∈𝐶%(𝐼) the solution of the Cauchy problem exists and is unique ´ In the classical Malthus equation 𝜀 is a constant, but we can generalize the solution for 𝜀(𝑡) with the generalized Malthus equation, which can be solved with separation of variables Separable variables When there are variables which can be separated, such as time and space variable, we can apply the separation of variables method: 𝑦!=𝐹(𝑡)∙𝐺(𝑦) Steps of resolution: i. Multiply both sides for dt: 𝑦!𝑑𝑡=𝐹(𝑡)∙𝐺(𝑦)𝑑𝑡 ii. Separate the variables under the condition 𝐺(𝑦)≠0: 𝑦!𝑑𝑡𝐺(𝑦)=𝐹(𝑡)𝑑𝑡 iii. Integrate and then apply the initial condition in order to find the value of the constant of integration: ∫𝑦!𝑑𝑡𝐺(𝑦)=∫𝐹(𝑡)𝑑𝑡 Generalized Malthus equation The generalized Malthus equation is characterized by a function 𝜀(𝑡) which multiply 𝑦 and depends on the variable 𝑡: 𝑦!𝑦=𝜀(𝑡) This equation can be solved with separation of variables Steps of resolution: i. Multiply both sides for dt: 𝑦!𝑦𝑑𝑡=𝜀(𝑡)𝑑𝑡 2 ii. Assume 𝑦!𝑑𝑡=𝑑𝑦 and integrate: ∫𝑑𝑦𝑦=∫𝜀(𝑡)𝑑𝑡 iii. Assume 𝐸!(𝑡)=𝜀(𝑡) to be the primitive of 𝜀(𝑡), so finally we obtain the solution: ln(𝑦)=𝐸(𝑡)+𝑘 → 𝑦=𝑒&($))* =𝐶𝑒&($) Non-separable variables There are also ODE in which the variables cannot be separated into…