Question

26 Find the solutions of the following initial-value problems: (a) cos(x + t)| _ + 1 | + 1 0, x(0) π 3 (b) 3(x + 2012ー+ 6(x +21)]/2 + 1 = 0. dt x(-1) 6

Answer 1

Any doubt then comment below..

These two problems are exact differential equatipn..first we prove that thiese equations is exact and then solve ..

a) cos (x + t) (dx/dt + 1) + 1 = 0 cos (x + 1) dx + (1 + cos (x + t)) dt = 0 M = cos (x + t) N = 1 + cos (x + t) partial differential M/partial differential d = -sin (x + t) partial differential N/partial differential x = -sin (x + t) partial differential M/partial differential y = partial differential N/partial differential x So, eq^n is exact General sol^n be integral M dx + integral (N not containing x) dt = C integral cos (x + 1) dx + integral 1 dt = C Sin (x + 1) + t = C x (0) = pi/2 So, sin (pi/2) + 0 = C C = 1 Sin (x + t) + t = 1 \r\n 3 (x + 2t)^1/2 dx/dt + 6 (x + 2t)^1/2 + 1 = 0 3 (x + 2t)^1/2