Consider x' + 2x = sin t
(a) Let x(t) and y(t) be two solutions. What is the upper bound on the separation |x(t) - y(t)| predicted by Theorem 7.15?
(b) Find the solution x(t) with initial value x(0) = -1/5 and the solution y(t) with initial value y(0) = -3/10. Does the separation x(t) - y(t) satisfy the inequality found in part (a)?
(c) Are there any values of t where the separation achieves the maximum predicted?
I see by inspection that a homogeneous solution is x = e^(-2t), thus the general solution should be x = e^(-2t) ∫sin(t)e^(2t) dt. Integrating by parts, I get x = (2sin(t) - cos(t) + Ce^(-2t))/5. I think the theorem I'm supposed to use is |x(t) - y(t)| ≤ |x0 - y0|e^(M|t - t0|), where M = max|∂f/∂x(t, x)|. I'm not sure where y(t) comes into the picture; are they saying that x(t) and y(t) represent two particular values of C, even though only x(t) appears in the general solution? ∂f/∂x(t, x) should be -2, but I'm not sure what else to do in part (a), which is prerequisite to part (b). I'm also not sure how to solve part (c); I assume I have to find M by somehow maximizing -2.
(a) Let x(t) and y(t) be two solutions. What is the upper bound on the separation |x(t) - y(t)| predicted by Theorem 7.15?
(b) Find the solution x(t) with initial value x(0) = -1/5 and the solution y(t) with initial value y(0) = -3/10. Does the separation x(t) - y(t) satisfy the inequality found in part (a)?
(c) Are there any values of t where the separation achieves the maximum predicted?
I see by inspection that a homogeneous solution is x = e^(-2t), thus the general solution should be x = e^(-2t) ∫sin(t)e^(2t) dt. Integrating by parts, I get x = (2sin(t) - cos(t) + Ce^(-2t))/5. I think the theorem I'm supposed to use is |x(t) - y(t)| ≤ |x0 - y0|e^(M|t - t0|), where M = max|∂f/∂x(t, x)|. I'm not sure where y(t) comes into the picture; are they saying that x(t) and y(t) represent two particular values of C, even though only x(t) appears in the general solution? ∂f/∂x(t, x) should be -2, but I'm not sure what else to do in part (a), which is prerequisite to part (b). I'm also not sure how to solve part (c); I assume I have to find M by somehow maximizing -2.