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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

3. Given solutions Y¡(t) = (x(1), yı(1)) and Y2(1) = (x2(t), y2(t)) to the system
dY
(a b)
AY,
where A =
dt
we define the Wronskian of Y (t) and Y2(t) to be the function
W(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 that
dW
= (a + d)W(t).
dt

(c) Find the general solution of the differential equation
dt
= (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) and
Y2(0) are linearly independent, then Y1(t) and Y2(1) are also linearly independent for every t.

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