Mathematics

Mathematics, 30.07.2019 21:20, lol15arec001

Consider the initial-value problem: dy/dt= 2te^(y-3), y(0) = 1. (a) show equation (1) possesses a unique local solution y(t). (b) compute the solution y(t) explicitly. (c) does the solution exist globally (for all times t)? why? (d) determine lim_t rightarrow -infinity y(t) and lim_t rightarrow infinity y(t) (if possible).

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Mathematics, 12.07.2019 20:30, eskarletche8

6. does the initial value problem tion in the neighborhood of x = 4 = 3y^2/3 where y(0) = 10-7 have a unique solu- 7. consider y 4 - 4x = 0. a) does theorem 1 imply the existence of a unique solution that satisfies y(xo) = 0? b) show that when xo + 0, there can not be a solution in any neighborhood of x = 20 such that y(xo) = 0. c) show that there are two distinct solutions to this ode satisfying y(0) = 0.

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Mathematics, 15.07.2019 18:20, Kingdcn6261

6. consider the following initial value problem: y 000 − y = 0 , y(0) = 1 , y0 (0) = 0 , y00(0) = 0 . (a) assume there is a solution of the form y(t) = e rt, and plug this back in and solve for r. use this to write the general solution. [hint: the equation r 3 = 1 has 3 distinct solutions over c, called the 3rd roots of unity: r1 = e 0πi/3 = 1, r2 = e 2πi/3 , and r2 = e 4πi/3 .] (b) from here, you could solve the initial value problem by plugging in the initial conditions, but you’ll get a system of three equations with three unknowns, and involving complex numbers. here’s another way: look for a power series solution, y(t) = x∞ n=0 ant n . plug this back into the ode and find the recurrence relation for the coefficients. (c) compute an for n ≤ 10, and once you see the pattern, write down the general solution to the ode as a power series. [hint: it should look familiar! ] (d) write down a basis for the solution space. (e) plug in the initial conditions and find the particular solution to the ivp. (f) solve a similar initial value problem:

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Consider the initial-value problem: dy/dt= 2te^(y-3), y(0) = 1. (a) show equation (1) possesses a u...

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