Recall that c₁y1 + c2y2 is called a fundamental set of solutions if it covers all possible solutions. (a) Verify that y₁(t) t² and y2(t) = = -1 t-1 are two solutions of the differential equation t²y" - 2y = 0 fort > 0. Then show that y = c₁t² + c₂t¹ is also a solution of this equation for any C1 and C2. Does this cover all the sets of solutions? (b) Consider the equation y" — y' — 2y = 0. -t - (b1). Show that y₁(t) = e¯t and y2(t) and y2(t) = e²t form a fundamental set of solutions. (b2). Let yз(t) 3 = —2e², y4(t) = y₁(t) +2y2(t), and y5(t) = 2y1(t) − 2y3(t). Determine whether each of the following pairs forms a fundamental set of solutions: {y2(t), y3(t)}; {y1(t), y4(t)}; {y4(t),y5(t)}.
Recall that c₁y1 + c2y2 is called a fundamental set of solutions if it covers all possible solutions. (a) Verify that y₁(t) t² and y2(t) = = -1 t-1 are two solutions of the differential equation t²y" - 2y = 0 fort > 0. Then show that y = c₁t² + c₂t¹ is also a solution of this equation for any C1 and C2. Does this cover all the sets of solutions? (b) Consider the equation y" — y' — 2y = 0. -t - (b1). Show that y₁(t) = e¯t and y2(t) and y2(t) = e²t form a fundamental set of solutions. (b2). Let yз(t) 3 = —2e², y4(t) = y₁(t) +2y2(t), and y5(t) = 2y1(t) − 2y3(t). Determine whether each of the following pairs forms a fundamental set of solutions: {y2(t), y3(t)}; {y1(t), y4(t)}; {y4(t),y5(t)}.
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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Transcribed Image Text:Recall that c₁y1 + c2y2 is called a fundamental set of solutions if it
covers all possible solutions.
(a) Verify that y₁(t) t² and y2(t)
=
=
-1
t-1 are two solutions of the differential equation
t²y" - 2y = 0 fort > 0. Then show that y = c₁t² + c₂t¹ is also a solution of this equation
for any C1 and C2. Does this cover all the sets of solutions?
(b) Consider the equation y" — y' — 2y = 0.
-t
-
(b1). Show that y₁(t) = e¯t and y2(t)
and y2(t) = e²t form a fundamental set of solutions.
(b2). Let yз(t)
3
=
—2e², y4(t) = y₁(t) +2y2(t), and y5(t) = 2y1(t) − 2y3(t). Determine
whether each of the following pairs forms a fundamental set of solutions:
{y2(t), y3(t)}; {y1(t), y4(t)}; {y4(t),y5(t)}.
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