Question

Determine an appropriate Green's funetion either by using the method of variation of parameters or otherwise, for the following boundary value problem: (b) 1+e

Answer 1

Given - (y^Prime (x) - y(x)) = 2/1 + e^-w y Prime (x) - y(x) = -2/1 + e^-x (D^2 - 1) y = -2/1 = e^-w A .E is f(x) = 0 = > m^2 - 1 = 0 = > m = -1 y_c = c_1 e^-x + c_2 e^x But given condition is not satisfy because y(0) = 0 = > 0 = c_1 e^-0 + c_2 e^0 0 = c_1 + c_2 = > c_2 = -c_1 y(1) = 1 = > 1 = c_1 e^-1 + c_2 e^1 so, It is not satisfy linear independent Near we assume that y_1(x) = sin hx put x = 0 = > y_1 (0) = sin h(0) = 0 & second condition y(1) = 0 = > y_2 (x) = sin h(x - 1) y_2 (1) = sin h(1 - 1) y_2 (1) = 0 The constant c is = y_1 y^1_2 - y_2 y^1_1 = c/p(x) Because L(y) = p y Prime + p prime y prime + qy = y Prime (Because p = 1)