DE 1

DE 1

1. Basic Terminology

   • A differential equation is a function of variables and their derivatives
   • The order of an ODE is the order of the highest derivatives
   • Dependent variables are unknown quantities, they depend on the equation. Dependent variables can usually be differentiated wrt to the independent variable(s), which do not determine on the equation. In x=f(t), t is the independent variable and x is dependent
     • If the equation has only one independent variable, it is called and ODE
   • If the independent variable does not appear explicitly in the ODE eg F(y,y'')=0, then it is called autonomous
   • An nth order ODE with the property of linearity is called linear, ie of the form:
     • • If f(t)=0, then it is called a linear homogenous equation
   • We can specify a particular solution by saying what F(x) is at a particular value of x, this is called an initial value problem

2. Existence and uniqueness

   • (*) Given an open interval, I, that contains to, a solution of the IVP: x'=f(x,t), x(t0)=x0 and x'(t)=f(x,t) for all t∈I. This means a solution does not have to be defined on all the reals, only an interval.
     • Thm of Existence and Uniqueness: If f(x,t) and ∂f/∂x(x,t) are cont. for a<x<b and for c<t<d then for and x0∈(a,b) t0∈(c,d), the initial value problem (*) has a unique solution on some open interval I containing t0.

3. Solving first order equations

   • Homogenous equations with constant coeffifients
     • dx/dt + px = 0 ⇒ x(t)=A*e^(-p*x)
   • Homogenous equations with non-constant coeffifients
     • dx/dt + p(t)x = q(t) Multiply both sides by an Integrating Factor P(t) = exp (int(p(t)dt)) so that d/dt(P(t)x(t)) = P(t)q(t)
       • • Note that y(t)=p(t)+h(t), where , p(t) is called the particular solution to the equation, while h(t) is the solution to the analogous homogenous case
   • Seperable equations
     • Consider dx/dt=f(x)g(t) where f is sufficiently nice Assume x(t) is a non-constant solution and hence f(x(t)) is non-zero on an interval I. Now divide both sides by f(x(t)): 1/f(x(t))*d/dt*x(t)=g(t)
       • Setting G(t) to the integral of g(t) and F(x) is an integral of 1/f(x) then we can integrate both sides and gain F(x(t))=G(t)+A
         • Note that this does not give an explicit solution for x(t), but rather a functional relationship between x(t) and t, also known as an implicit solution. We would like to rearrange this to get an explicit solution but this is not always possible

4. Substitution Methods

   • dy/dx=F(y/x) use the substitution y=ux to gain a seperable equation
   • Equations of the form: dy/dx + p(x)y=q(x)y^n are knowen as Bernoulli equations
     • for n>1 we use the substitution u=y^(n-1)

5. Direction Fields

   • In the enlarged phase space is the space of x vs. t, in which every point has a vector associated with is having f(x,t) with sometimes the length proportional to |f(x,t)|. All vectors form the "direction fields"
   • For an initial value, if we draw the solution to equation through the initial value, it is tangental to every vector. Only one curve goes through any one point due to uniqueness. Such curves are caled integral curves

6. Autonomous First-Order ODEs

   • We may not always be able to find an explicit solution to an ODE. One class is dx/dt=f(x(t))
     • Then we can graph dx/dt=f(x(t)), and mark arrows describing the direction a particle would move in as t varies
       • At dx/dt(x)=0 we have fixed points
         • dx/dt→∞ is called asymptotic behaviour
         • If a particle is drawn to a fixed point it is "stable", while if it is repelled it is "unstable"
           • f'(x)<0⇒stable, f'(x)>0⇒unstable,
             • If f'(x)=0 at a fixed point, the equation is called structurally unstable, a small change to the equation: dx/dt=f(x)+ε(x): |ε(x)|<<1
               • Can make the fixed point stable or unstable by making f'(x) non-zero at the fixed point

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