4. [18 marks] Solve for u(x, t) and, using the result from Part 2, show that v(x, t) = En-1 bnw(t)un(x, t) where you should find un(x, t). HINT: This is subtly dif- ferent to the PDEs you have seen in lectures but the procedure of separation of variables works in this case since (5) is linear and homogeneous. 5. [15 marks] Show that if kL? (1] l > (L G > U, and C > U are constants describing the electrical properties of the cable and can be adjusted by clever manufacturing of the cable but never set to zero. 1. [11 marks] Eliminate 1', to show how (3) reduces to the 'Telegrapher's equation' 8211 (97.: 2 821.: w'l'g'l'b'UC, (4) by describing the constants a, b and c in terms Of R, l, G and C. 2. [12 marks] Rewrite BUS, t) : m(t)u(;r. t) and choose a suitable nr(t) such that the Teleg- rapher's equation simplifies to alt opt 0. 4. U + + t ku = cu xx The ulx, of = > businc max ) (t ( x, 0 ) = 0 n = 1 try to find solution in the form X T " + R X T = C X T C ' T = * " = Klconstant ) D X " - K X = 0 ( T " + K T = K C ' T T = KC'T- RT I" = ( KC ' - k ) T I' - ( KC" - K ) T = 0 I " + 6 k - KC ' ) T = 0 U l o , t ) = X cos Tut ) = 0 2 5 X() = D " ( L , t ) = X CL ) Tut ) = 0 1 X LL ) = 0 U t ( x , 0 ) = X ( X ) T ( 0 ) = 0 - T ( ) = D D X " - K X = 0 X 1 ) = 0 X CL ) = 0 *