Mathematics

Mathematics, 17.07.2019 22:20, jalayshiamartin

Asecond order ivp and its solution are given below. differentiate the solution and then plug initial conditions into the solution and its derivative to generate a system of two linear equations in the two unknowns c_1 and c_2. solve the system for c_1 and c_2. y" - 25y = 0 y(0) = 10 y'(0) = 20 y(x) =20 y(x) = c_1e^5x + c_2 e^-5x

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Mathematics, 04.07.2019 11:50, GamerGirl15

1point) the solution of a certain differential equation is of the form y(t)=aexp(3t)+bexp(4t), y ( t ) = a exp ⁡ ( 3 t ) + b exp ⁡ ( 4 t ) , where a a and b b are constants. the solution has initial conditions y(0)=3 y ( 0 ) = 3 and y′(0)=3. y ′ ( 0 ) = 3 . find the solution by using the initial conditions to get linear equations for a a and

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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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Mathematics, 30.07.2019 19:10, chrismcb875

solve using greens theorem: we seek to solve: d^ 2 y/ dx^2 + 4y = 4 sec 2x tan 2x, subject to the initial conditions y(0) = 3 , y'(0) = −4 (a) find the solution to the homogeneous equation with the nonhomogeneous boundary conditions (yh). (b) what is the green’s function for this operator? (c) what is the particular solution (yp)? (d) what is the solution to the initial value problem ((1)& ( (e) what is the largest interval starting at x = 0 for which the solution is valid?

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Mathematics, 02.08.2019 17:20, kell22wolf

Gconsider the following b/ivp for the wave equation, defined for 0 ≤ x ≤ 1 and t ≥ 0: utt = c 2uxx, u(0, t) = u(1, t) = 0, u(x, 0) = x(1 − x), ut(x, 0) = 0. (a) briefly describe, and sketch, a physical situation which this models. be sure to explain the effect of both boundary conditions and both initial conditions. (b) assume that u(x, t) = f(x)g(t). find utt and uxx. also, determine the boundary conditions for f(x) from the boundary conditions for u(x, t), and one initial condition for g(t). (the three bc/ics equal to zero will give you these.) (c) plug u = fg back into the pde and divide both sides by c 2 fg, i.e., “separate variables.” briefly justify why this quantity must be a constant. call this constant −λ. write down two odes: one for g(t) with one initial condition, and a sturmlioville problem for f(x). (d) determine f(x), λ, and g(t). (e) using your solution to part (d) and the principle of superposition, find the general solution to this pde. (f) use the last initial condition, u(x, 0) = x(1 − x), to find the particular solution that satisfies the initial and boundary conditions. (g) what is the long-term behavior of this system? give both a mathematical and physical justification.

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