Question

4. Solve ul inx COS су where u( π/2,"y-y(y +1).

*Note: Please kist all steps and explain all workings. Thank you!

Answer 1

Sin x partial y/partial x + cos x partial y/partial y = cos x So, dx/sinx = dy/cos x = dy/cos x therefore cot x dx = dy implies u mod sin x mod = y_c_1 (constant) therefore sin x = A e^y , A = e^e1 and, dy = du implies u = y+ B implies u = y + f(sinx/e^y) , B = f(A) Now, u(phi/2, y) = y(y+ 1) therefore y(y+ 1) = y + f(1/e^y) implies y^2 = f(e^-y) therefore f(y) = 1(ln y)^2, u(x,y) = y + 1/[ln(sinx/e^y)]^2 (Ans)