3. Suppose that the solution of a one-dimensional wave equation is given by u(x,t) = F(x-ct) + G(x + ct), where 0, x <0 0, x < 0 F(x) = 2, 0 < x <2 and G(x)= x, 0 < x <1 0, x 2 0, x > 1. Sketch the solution for various times.

4. Consider a wave equation on an infinite line, J²u J²u 9 Ət² əx² = 0. = Find the characteristics though the point (1,3). Draw the domains of depen- dence and influence of the point (1,3).

Q.1) A- If x=u-vand y = u + v then show the equations that correspond to Əw dw Əw aw. the following equation: (4)2 (44) 2 = dy dx Ou Əv A

4. Find the parametric equation of the line through (1, 0, -2) and parallel to the line %(x-4) = ½ (y) = z+ 2.

6. Find the image equation of the line –2x + y + 3 = 0 under the rotation about C = (1, 1) at an angle e = 90°.

Which of the following is most suitable solution for wave equation = c2? %3D at2 ax2 .(a) y = (AeP* + Be-P*)(Ce pt + De ept) (b) y = (Acos px + Bsin px)(Ccos cpt + Dsin cpt) (c) y = (Ax + B)(Ct + D) (d) y = (Aepx + Be-px)(Cep*t + De-ep*r) O a O b O c O d

2. Sketch the level curve z = k for the indicated values of k.