1.1:Examples of differentiable equation:1.) F = ma = mdvdt= mg − γv2.) Mouse population increases at a rate proportionalto the current population:More general model :dpdt= rp − kwhere r = growth rate or rate constant,k = predation rate = # mice killed per unit time.direction field = slope field = graph ofdvdtin t, v-plane.*** can use slope field to determine behavior of vincluding as t → ∞.Equilibrium Solution = constant solution11.2:Solveddydt= ay + b by separating variables:dyay+b= dtRdyay+b=Rdtln|ay+b|a= t + Cln|ay + b| = at + Celn|ay+b|= eat+C|ay + b| = eCeatay + b = ±(eCeat)ay = Ceat− by = Ceat−baInitial Value Problem: y(t0) = y021.3:ODE (ordinary differential equation): single indepen-dent variableEx:dydt= ay + bvsPDE (partial differential equation): sever al indepen-dent variablesEx:∂xy∂x=∂xy∂yorder of differential eq’n: order of highest derivativeexample of order n : y(n)= f(t, y, ..., y(n−1))Linear vs Non-linearlinear: a0(t)y(n)+ ... + an(t)y = g(t)Ex: ty′′− t3y′− 3y = sin(t)Ex: 2y′′− 3y′− 3y2= 03********Existence of a solution**********************Uniqueness of solution***************CH 2: Solvedydt= f(t, y)2.1: First order linear eqn:dydt+ p(t)y = g(t)Ex 1: y′= ay + bEx 2: y′+ 3t2y = t2, y(0) = 0Note: could use section 2.2 method, separationvariables to solve ex 1 and 2.Ex 3: t2y′+ 2ty = tsin(t)Ex 1: 2dydt+ 10y =
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