Introduction to Ordinary Differential Equations Emily Weymier Department of Mathematics & Statistics Stephen F. Austin State University, Nacogdoches, TX September 22, 2017 Emily Weymier (SFA) Perturbation Theory September 22, 2017 1 / 28
Outline 1 What is a differential equation? 2 Initial Value Problems Linear first order differential equations Second order differential equations Recasting high order differential equations as a system of first order differential equations 3 Boundary Value Problems 4 Solution techniques for nonlinear differential equations Power series solutions Perturbation theory concept 5 Concluding Remarks Emily Weymier (SFA) Perturbation Theory September 22, 2017 2 / 28
Differential Equations: The Basics Ordinary differential equations are used to model change over an independent variable (for our purposes it will usually be t for time) without using partial derivatives. Emily Weymier (SFA) Perturbation Theory September 22, 2017 3 / 28
Differential Equations: The Basics Ordinary differential equations are used to model change over an independent variable (for our purposes it will usually be t for time) without using partial derivatives. Differential equations contain three types of variables: an independent variable, at least one dependent variable (these will be functions of the independent variable), and the parameters. Emily Weymier (SFA) Perturbation Theory September 22, 2017 3 / 28
Differential Equations: The Basics Ordinary differential equations are used to model change over an independent variable (for our purposes it will usually be t for time) without using partial derivatives. Differential equations contain three types of variables: an independent variable, at least one dependent variable (these will be functions of the independent variable), and the parameters. ODE’s can contain multiple iterations of derivatives. They are named accordingly (i.e. if there are only first derivatives, then the ODE is called a first order ODE). Emily Weymier (SFA) Perturbation Theory September 22, 2017 3 / 28
Differential Equations: The Basics Ordinary differential equations are used to model change over an independent variable (for our purposes it will usually be t for time) without using partial derivatives. Differential equations contain three types of variables: an independent variable, at least one dependent variable (these will be functions of the independent variable), and the parameters. ODE’s can contain multiple iterations of derivatives. They are named accordingly (i.e. if there are only first derivatives, then the ODE is called a first order ODE). Emily Weymier (SFA) Perturbation Theory September 22, 2017 3 / 28
A Simple Example: Population Modeling Population growth is commonly modeled with differential equations. In the following equation: t = time, P = population and k = proportionality constant. k represents the constant ratio between the growth rate of the population and the size of the population. dP dt = kP In this particular equation, the left hand side represents the growth rate of the population being proportional to the size of the population P . This is a very simple example of a first order, ordinary differential equation. The equation only contains first order derivatives and there are no partial derivatives. Emily Weymier (SFA) Perturbation Theory September 22, 2017 4 / 28
Initial Value Problems An initial value problem consists of a differential equation and an initial condition. So, going back to the population example, the following is an example of an initial value problem: dP dt = kP , P (0) = P 0 The solution to this set of equations is a function, call it P ( t ), that satisfies both equations. Emily Weymier (SFA) Perturbation Theory September 22, 2017 5 / 28
Linear First Order Differential Equations The standard form for a first-order differential equation is dy dt = f ( t , y ) where the right hand side represents the function f that depends on the independent variable, t , and the dependent variable, y . Emily Weymier (SFA) Perturbation Theory September 22, 2017 6 / 28
General Solutions to a Differential Equation Let’s look at a simple example and walk through the steps of finding a general solution to the following equation dy dt = ( ty ) 2 We will simply ”separate” write as “separate” the variables then integrate the both sides of the equation to find the general solution. dy t 2 y 2 = dt 1 t 2 dt y 2 dy = � 1 � t 2 dt y 2 dy = Emily Weymier (SFA) Perturbation Theory September 22, 2017 7 / 28
t 3 − y − 1 = 3 + c t 3 − 1 = 3 + c y 1 = − y t 3 3 + c 3 ⇒ y ( t ) = − t 3 + c 1 where c 1 is any real number. Emily Weymier (SFA) Perturbation Theory September 22, 2017 8 / 28
Linear First Order Differential Equations Initial value problems consist of a differential equation and an initial value. We will work through the example below: dx 1 dt = − xt ; x (0) = √ π First we will need to find the general solution to dx dt = − xt , then use the 1 initial value x (0) = √ π to solve for c . Since we do not know what x ( t ) is, we will need to ”separate” the equation before integrating. dx = − xt dt − 1 = x dx t dt � − 1 � x dx = t dt Emily Weymier (SFA) Perturbation Theory September 22, 2017 9 / 28
Linear First Order Differential Equations Continued t 2 − ln x = 2 + c e − ( t 2 2 + c ) = x e − ( t 2 2 ) e − c = x ke − t 2 x = 2 The above function of t is the general solution to dx dt = − xt where k is 1 some constant. Since we have the initial value x (0) = √ π , we can solve for k . Emily Weymier (SFA) Perturbation Theory September 22, 2017 10 / 28
Solving Initial Value Problems Thus we can see that the solution to the initial value problem 1 dx dt = − xt ; x (0) = √ π is 1 √ π = ke − 02 x (0) = 2 1 √ π e − t 2 x ( t ) = 2 Emily Weymier (SFA) Perturbation Theory September 22, 2017 11 / 28
Let’s verify that this solution is correct. We will need to show dx x ′ ( t ) = f ( t , x ( t )) = dt � 1 � 1 dx d √ π e − t 2 √ π e − t 2 = = 2 2 dt dt � 1 � = − 1 d √ π e − t 2 √ π e − t 2 ⇒ 2 2 dt Emily Weymier (SFA) Perturbation Theory September 22, 2017 12 / 28
Second Order Differential Equations Second order differential equations simply have a second derivative of the dependent variable. The following is a common example that models a simple harmonic oscillator: d 2 y dt 2 + k my = 0 where m and k are determined by the mass and spring involved. This second order differential equation can be rewritten as the following first order differential equation: dv dt = − k my where v denotes velocity. Emily Weymier (SFA) Perturbation Theory September 22, 2017 13 / 28
Second Order Differential Equations Continued Referring back to some calculus knowledge, if v ( t ) is velocity, then v = dy dt . Thus, we can substitute in dv dt into our second order differential equation and essentially turn it into a first order differential equation. d 2 y dt 2 = − k my ⇔ dv dt = − k my Now we have the following system of first order differential equations to describe the original second order differential equation: dy = v dt dv − k = my dt Emily Weymier (SFA) Perturbation Theory September 22, 2017 14 / 28
Second Order Differential Equations Continued Consider the following initial value problem: d 2 y dt 2 + y = 0 with y (0) = 0 and y ′ (0) = v (0) = 1. Let’s show that y ( t ) = sin( t ) is a solution. Let v = dy dt , then we have the following system: dy = v dt dv = − y dt Emily Weymier (SFA) Perturbation Theory September 22, 2017 15 / 28
Second Order Differential Equations Continued dy d = dt sin( t ) = cos( t ) = v dt dv = − sin( t ) = − y dt ⇒ d 2 y = − sin( t ) dt 2 ⇒ d 2 y d 2 (sin( t )) dt 2 + y = + sin( t ) dt 2 = − sin( t ) + sin( t ) = 0 Emily Weymier (SFA) Perturbation Theory September 22, 2017 16 / 28
High Order Differential Equations as a System Emily Weymier (SFA) Perturbation Theory September 22, 2017 17 / 28
Boundary Value Problems: The Basics Emily Weymier (SFA) Perturbation Theory September 22, 2017 18 / 28
Power Series Solutions To demonstrate how to use power series to solve a nonlinear differential equation we will look at Hermite’s Equation: d 2 y dt 2 − 2 t dy dt + 2 py = 0 We will use the following power series and its first and second derivatives to make a guess: ∞ a 0 + a 1 t + a 2 t 2 + a 3 t 3 + ... = � a n t n y ( t ) = (1) n =0 ∞ dy a 1 + 2 a 2 t + 3 a 3 t 2 + 4 a 4 t 3 + ... = � na n t n − 1 = (2) dt n =1 ∞ d 2 y 2 a 2 + 6 a 3 t + 12 a 4 t 2 + ... = � n ( n − 1) a n t n − 2 = (3) dt 2 n =2 Emily Weymier (SFA) Perturbation Theory September 22, 2017 19 / 28
From the previous equations we can conclude that y (0) = a 0 y ′ (0) = a 1 Next we will substitute (1), (2) and (3) into Hermite’s Equation and collect like terms. d 2 y dt 2 − 2 t dy dt + 2 py = 0 = (2 a 2 + 6 a 3 t + 12 a 4 t 2 + ... ) − 2 t ( a 1 + 2 a 2 t + 3 a 3 t 2 + 4 a 4 t 3 + ... ) +2 p ( a 0 + a 1 t + a 2 t 2 + a 3 t 3 + ... ) ⇒ (2 pa 0 + 2 a 2 ) + (2 pa 1 − 2 a 1 + 6 a 3 ) t + (2 pa 2 − 4 a 2 + 12 a 4 ) t 2 + (2 pa 3 − 6 a 3 + 20 a 5 ) t 3 = 0 Emily Weymier (SFA) Perturbation Theory September 22, 2017 20 / 28
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