Consider the first order ordinary differential equation Find the general solutio

Question

Consider the first order ordinary differential equation Find the general solution for 0 t 2 [5 marks]. Find the constant in the general solution for 0 t 2 so that the initial condition is satisfied [2 marks]. Find a general solution for t > 2 [5 marks]. Find the constant in the general solution for t > 2 so that the solution for t > 2 and the solution for 0 t 2 match at t = 2 [5 marks]. Use Maple, Matlab, or MS-excel to plot the solution for 0 t 5 [3 marks]. Explain, with reference to your expression for y(t) and plot of y(t), whether or not the solution is continuous at t = 2 [1 mark]. Explain, with reference to your expression for y(t) and plot of y(t), whether or not the solution is differentiable at t = 2 [1 mark].

Explanation / Answer

Let p(t) and g(t) be continuous functions on the open interval (a, b) and let t0 be a point in (a, b). Then the initial value problem dy/dt+ p(t)y = g(t), y(t0) = y0 3has a unique solution on the entire interval (a, b). Our goal is to be able to integrate both sides, and thereby eliminate the derivative. We note that dy/dt + p(t)y looks a little bit like a product rule, if the derivative of one term produces an extra p(t). So we multiply through the entire equation by e^f(t) where f?(t) = p(t), so f(t) = ?p(t) dt. This exponential function is what we call an integrating factor. (Since after all it is a factor which helps us integrate the left hand side of the equation.) Thus, we start by multiplying both sides by our integrating factor exp (? p(t) dt) dy/dt+ exp (? p (t) dt) p(t)y = exp (? p(t) dt) g(t) (Recall that exp(x) is an alternate way of writing e^x Then we have as before [exp (?p(t) dt) y] = g(t) exp (?p(t) dt), which (upon integrating) gives exp (?p(t) dt) y = ?g(t) exp (?p(t) dt) dt. Solving for y then results in the solution: y = exp(??p(t) dt) exp (?p(t) dt) g(t) dt This of course involves two di?erent integrations! Note that the integral will of course include some arbitrary constant (

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