2. This problem will illustrate a technique for solving the homogeneous linear differential equation y" = p(t)y' +q(t)y.

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2 This Problem Will Illustrate A Technique For Solving The Homogeneous Linear Differential Equation Y P T Y Q T Y 1 (113.77 KiB) Viewed 17 times

2. This problem will illustrate a technique for solving the homogeneous linear differential equation y" = p(t)y' +q(t)y. Suppose that y₁ (t) and y2(t) are two solutions to such an equation. (a) Show that the Wronskian W(t) of y₁ and y2 satisfies the differential equation W'(t) = p(t)W(t). Deduce that W(t) = CeP(t) for some constant C, where P(t) is a choice of anti-derivative of p(t). 3/2 wt). This means that (b) Show that y₁ satisfies the differential equation y' if one solution, say y2, is known, you can solve a first order ÖDE to find another one. (c) Suppose I tell you that y2(t) = t² is a solution to y" = y + y. Use the method outlined in the previous parts to find a second solution y₁(t). (Hint: you can take the constant of integration when you solve the first order ODE to be 0, since you're only interested in one solution!) (d) Explain why your solutions y₁ and y2 are linearly independent, and write down the general solution to the second order ODE in part (c).