3. Given solutions Y¡(t) = (x(1), yı(1)) and Y2(1) = (x2(t), y2(t)) to the systemdY(a b)AY,where A =dtwe define the Wronskian of Y (t) and Y2(t) to be the functionW(t) = x1(1)y2(1) – x2(1)y1(1).(a) Compute dW/dt.(b) Use the fact that Y1(1) and Y2(1) are solutions of the linear system to show thatdW= (a + d)W(t).dt (c) Find the general solution of the differential equationdt= (a + d)W(t).(d) Suppose that Y¡(t) and Y2(t) are solutions to the system dY/dt = AY. Verify that if Y¡(0) andY2(0) are linearly independent, then Y1(t) and Y2(1) are also linearly independent for every t.

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Given solutions Y¡(t) = (x1(t), yı(t)) and Y2(t) = (x2(t), y2(t)) to the system dY a b = AY, where A=| dt c d we define the Wronskian of Y1(1) and Y2(t) to be the function W(t) = x1(1)y2(t) – x2(1)y1(1). (a) Compute dW/dt. (b) Use the fact that Y1(1) and Y2(1) are solutions of the linear system to show that dW = (a + d)W(t). dt

Given solutions Y¡(t) = (x1(t), yı(t)) and Y2(t) = (x2(t), y2(t)) to the system dY a b = AY, where A=| dt c d we define the Wronskian of Y1(1) and Y2(t) to be the function W(t) = x1(1)y2(t) – x2(1)y1(1). (a) Compute dW/dt. (b) Use the fact that Y1(1) and Y2(1) are solutions of the linear system to show that dW = (a + d)W(t). dt

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...

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