University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2021
Views:

79

Pages:

2
Author:

Anthony Bauer

Quiz 5 SOLUTIONS 1. True or False: The fundamental set for a given second order, linear, homogeneous differential equation is unique. If this statement is false, give a specific example. If the statement is true, say why (e.g., state a theorem): False- Any set of functions can be a fundamental set, as long as they are two linearly independent solutions. For example, if y1 and y2 form a fundamental set, so do: y1 + y2 y1 − y2 The idea here is that any set of 2 linearly independent functions that are solutions can be used to form a fundamental set- In particular, we just need to make sure that the Wronskian at t0 is not zero. However, a set like y1 + y2 and 2y1 + 2y2 would not make up a fundamental set. 2. Give the relationship between the Wronskian of two functions and the linear (in)dependence of functions by filling in the following: • If W (f, g)(t0 ) 6= 0, then: f, g are linearly independent on any interval containing t0 . • If f, g are linearly dependent on an open interval I, then: W (f, g)(t) = 0 for all t in the interval I. • If W (f, g)(t0 ) = 0 for all t in an open interval I, then: (Hint- two cases to consider) – If f, g are two functions in general, we cannot conclude anything about the linear (in)dependence. – If f, g are two solutions to the linear, second order differential equation where p, q are continuous, then f, g are linearly dependent on I (By Abel) Side Remark: If f, g are linearly independent on an interval I, W (f, g)(t) could be zero for all t in I. 3. Find the Wronskian of two solutions to the following ODE without solving the DE: (1 − x2 )y 00 − 2xy 0 + 3y = 0 − W (y1 , y2 )(t) = Ce R p(t) dt where p(t) = −2x/(1 − x2 ). Integrating, − Z − Z 1 2x dx = − du = − ln |1 − x2 | 1 − x2 u Therefore, 2 W (y1 , y2 ) = Ce− ln |1−x | = C 1 − x2 4. Give the general solution to the ODE, where the following are the associated characteristic equations: 1 (a) (r − 1)(r + 2) = 0 (b) (r − 1)2 = 0 (c) r2 + 4 = 0 y(t) = C1 et + C2 e−2t y(t) = et (C1 + C2 t) y(t) = C1 cos(2t) + C2 sin(2t) 5. For each characteristic equation given in (4), give the differential equation that you solved: (a) y 00 + y 0 − 2y = 0 (b) y 00 − 2y 0 + y = 0 (c) y 00 + 4y = 0 2

Differential Equations Quiz 5 Solutions

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