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Mathematics · 400 Level
This is the complete lecture note for MTH 402 Theory Of Partial Differential Equations, prepared for Mathematics students at 400 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1: First Order and Second Order Linear Equations Series Outline Part 1.1: Introduction to First Order Linear Equations Definition and general form Methods of solution (separable, homogeneous, integrating factor) Simple applications Part 1.2: Second Order Linear Equations Definition and general form Homogeneous equations with constant coefficients Particular solutions and general solutions Part 1.3: Canonical Forms and Solution Techniques Reduction to canonical forms Solution strategies for different cases Worked examples Part 1.4: Applications of First and Second Order Linear Equations Physical and engineering contexts Practice problems Introduction Partial differential equations often begin with the study of simpler cases, namely first order and second order linear equations. These equations form the foundation for more advanced topics, since they capture essential behaviours such as growth, decay, oscillation, and equilibrium. By mastering these, you will be well prepared to tackle more complex PDEs later in the course. The aim of this Series is to introduce you to the theory and solution methods of first order and second order linear equations, and to equip you with the skills to apply these methods in practical contexts. We shall commence this Series by defining first order linear equations and exploring their solution methods. Next, we will move on to second order linear equations, examining both homogeneous and particular solutions. Following that, we will consider canonical forms and solution techniques, supported by worked
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS examples. Finally, we will conclude with applications of these equations in physical and engineering contexts, rounding off with practice problems to consolidate your learning. Series Learning Outcomes Upon completing this Series, you should be able to: SLO 1.1 define first order linear equations and explain their general form, SLO 1.2 demonstrate methods of solving first order linear equations, including separable, homogeneous, and integrating factor techniques, SLO 1.3 define second order linear equations and explain their general form, SLO 1.4 apply solution methods to homogeneous second order linear equations with constant coefficients, SLO 1.5 analyse canonical forms of linear equations and evaluate solution strategies, and SLO 1.6 apply first and second order linear equations to physical and engineering problems. 1.1 Introduction to First Order Linear Equations First order linear equations are the simplest type of partial differential equations and provide the foundation for more advanced study. A first order linear equation is an equation that involves the first derivative of a function but no higher derivatives. Its general form is: dydx+P(x)y=Q(x) where P(x)and Q(x)are functions of x. These equations are important because they describe basic processes such as exponential growth, decay, and equilibrium. Fill in the blank: A first order linear equation involves the __________ derivative of a function but no higher derivatives. 1.1.1 Definition and general form The general form of a first order linear equation can be recognised by the presence of the first derivative and a linear dependence on the unknown function y(x). The equation does not contain terms like y2or (dy/dx)2, which would make it nonlinear.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS General form of a first order linear equation. Fill in the blank: The general form of a first order linear equation is \frac{dy}{dx} + P(x)y = __________. 1.1.2 Methods of solution There are several methods for solving first order linear equations: Separable equations: These are equations where variables can be separated on different sides of the equation. Homogeneous equations: These equations can be simplified by substitution, often reducing them to separable form. Integrating factor method: This is a systematic technique where the equation is multiplied by a specially chosen function to make the left-hand side an exact derivative. Methods of solving first order linear equations Separable equations
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Homogeneous equations Integrating factor method Fill in the blank: The integrating factor method involves multiplying the equation by a specially chosen __________ to make it easier to solve. 1.1.3 Simple applications First order linear equations appear in many physical and engineering contexts. For example: Exponential growth and decay: Population growth or radioactive decay. Cooling laws: Newton’s law of cooling. Electrical circuits: Current and voltage relationships in resistive circuits. Applications of first order linear equations Application Equation form Notes Population growth dydt=ky Exponential growth Radioactive decay dydt=-ky Exponential decay Newton’s cooling law dTdt=-k(T-Tenv) Cooling towards environment Fill in the blank: Newton’s law of cooling is modelled by a first order linear equation where temperature approaches the __________ temperature. Pilot questions 1.1
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS A first order linear equation involves: A. The first derivative only B. The second derivative only C. Both first and second derivatives D. None of the above (SLO 1.1) The general form of a first order linear equation is: A. dydx+P(x)y=Q(x)B. dydx+y2=Q(x)C. dydx+P(x)y2=Q(x)D. None of the above (SLO 1.1) Which method involves multiplying the equation by a specially chosen function? A. Separable method B. Homogeneous method C. Integrating factor method D. None of the above (SLO 1.2) Newton’s law of cooling is an example of: A. A first order linear equation B. A second order linear equation C. A nonlinear equation D. None of the above (SLO 1.6) Radioactive decay is modelled by: A. dydt=kyB. dydt=-kyC. dydt=y2D. None of the above (SLO 1.6) 1.2 Second Order Linear Equations Second order linear equations are a step up in complexity from first order equations, yet they are equally fundamental in the study of partial differential equations. A second order linear equation is one that involves the second derivative of a function, with the unknown function and its derivatives appearing linearly. The general form is: a(x)d2ydx2+b(x)dydx+c(x)y=f(x) where a(x), b(x), and c(x)are functions of x, and f(x)is a given function. Fill in the blank: A second order linear equation involves the __________ derivative of a function, along with lower order terms. 1.2.1 Definition and general form The defining feature of second order linear equations is the presence of the second derivative. These equations are linear because the unknown function y(x)and its derivatives appear only to the first power and are not multiplied together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS General form of a second order linear equation. Fill in the blank: The general form of a second order linear equation is a(x)y'' + b(x)y' + c(x)y = __________. 1.2.2 Homogeneous equations with constant coefficients A homogeneous equation is one where f(x)=0. If the coefficients a, b, and care constants, the equation takes the form: ay''+by'+cy=0 The solution depends on the characteristic equation: ar2+br+c=0 If the roots are real and distinct, the solution is a combination of exponentials. If the roots are real and equal, the solution includes a term with x. If the roots are complex, the solution involves sine and cosine functions. Solutions of homogeneous second order equations with constant coefficients
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Roots of characteristic equation General solution Real and distinct y=C1er1x+C2er2x Real and equal y=(C1+C2x)erx Complex y=eαx(C1cosβx+C2sinβx) Fill in the blank: If the characteristic equation has complex roots, the solution involves __________ and cosine functions. 1.2.3 Particular solutions and general solutions For non-homogeneous equations where f(x)0, the solution is the sum of: The complementary function (CF), which solves the homogeneous equation. The particular integral (PI), which is a specific solution to the non-homogeneous equation. Thus, the general solution is: y(x)=CF+PI Different methods such as undetermined coefficients or variation of parameters can be used to find the particular integral. Structure of solutions to second order linear equations Complementary function (CF): solution to homogeneous equation Particular integral (PI): solution to non-homogeneous equation General solution: sum of CF and PI
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