then Show that if u(x,t) = F(x + ct) + G(x - ct) satisfies u(x, 0)...

Transcribed Image Text:

then Show that if u(x,t) = F(x + ct) + G(x - ct) satisfies u(x, 0) = f(x), u(x, 0) = g(x), for all x = R, 1 C 2c f(x) 9(x)dx ((x)) = (-1) (())- Exercise 3. Continuing the notation of the preceding exercise, if G₁(x) and G2(x) are both antiderivatives of g(x), show that u(x, t) does not depend on which one we choose to represent g(x) dx. [Suggestion: Write down an equation relating G₁ and G2.] Exercise 4. Use exercises 2 and 3 to help you solve the 1-D wave equation (on the domain Rx [0, ∞)) subject to the given initial data. a. u(x, 0) = f(x), u(x, 0) = 0 1 b. u(x, 0) = 1+x²' ut(x, 0) = -2xe-x² == c. u(x, 0) = ex², ut(x, 0) = = (1 + x2)2 X then Show that if u(x,t) = F(x + ct) + G(x - ct) satisfies u(x, 0) = f(x), u(x, 0) = g(x), for all x = R, 1 C 2c f(x) 9(x)dx ((x)) = (-1) (())- Exercise 3. Continuing the notation of the preceding exercise, if G₁(x) and G2(x) are both antiderivatives of g(x), show that u(x, t) does not depend on which one we choose to represent g(x) dx. [Suggestion: Write down an equation relating G₁ and G2.] Exercise 4. Use exercises 2 and 3 to help you solve the 1-D wave equation (on the domain Rx [0, ∞)) subject to the given initial data. a. u(x, 0) = f(x), u(x, 0) = 0 1 b. u(x, 0) = 1+x²' ut(x, 0) = -2xe-x² == c. u(x, 0) = ex², ut(x, 0) = = (1 + x2)2 X

Answer rating: 100% (QA)

To solve these exercises lets start with Exercise 3 and then move on to Exercise 2 After that well use the results to solve Exercise 4 Exercise 3 We h... View the full answer