Problem 1. Consider the telegraph equation a2u at2 6202 cxu. a.r2 This is a partial differential equation that models the voltage or current on an electrical transmission line. Verify that u(x, t) Jo(As)f(y) dy 2c is a solution to the telegraph equation. Here, c242 (x y)?, S = and Jn is the Bessel function of the first kind defined by 2 Jn (2) = so cos(z sin 0 - no) do. 7T Hints: Chain rule. Chain rule. Chain rule. You may find the Leibniz integration rule to be useful (itself a consequence of the chain rule and the fundamental theorem of Calculus): a pb(,t) a p(at) ab(x, t) g(t, x,y) dy = g(t, x, b(x, t)) at Da(.x, t) g(t, x, a(x, t)) at + at Ja(0,t) Jacz;t) 379(t, x, y) dy po(t) on Saterem g(t, x,y) dy ab(x, t) = g(t, x, b(.x, t)) aa(x, t) g(t, x, a(r, t)) + Selecteer opg(0,29) day. Also, you may want to use the following properties of Bessel functions of the first kind: Jo(0) =1, J10) = 0 J(z) = -J1(2), 3(z) = J.(2) J1(2) Problem 1. Consider the telegraph equation a2u at2 6202 cxu. a.r2 This is a partial differential equation that models the voltage or current on an electrical transmission line. Verify that u(x, t) Jo(As)f(y) dy 2c is a solution to the telegraph equation. Here, c242 (x y)?, S = and Jn is the Bessel function of the first kind defined by 2 Jn (2) = so cos(z sin 0 - no) do. 7T Hints: Chain rule. Chain rule. Chain rule. You may find the Leibniz integration rule to be useful (itself a consequence of the chain rule and the fundamental theorem of Calculus): a pb(,t) a p(at) ab(x, t) g(t, x,y) dy = g(t, x, b(x, t)) at Da(.x, t) g(t, x, a(x, t)) at + at Ja(0,t) Jacz;t) 379(t, x, y) dy po(t) on Saterem g(t, x,y) dy ab(x, t) = g(t, x, b(.x, t)) aa(x, t) g(t, x, a(r, t)) + Selecteer opg(0,29) day. Also, you may want to use the following properties of Bessel functions of the first kind: Jo(0) =1, J10) = 0 J(z) = -J1(2), 3(z) = J.(2) J1(2)