MT1710: Question Sheet 6 To be handed in: Tuesday 13 November 2018 (before 1pm, in the Maths office, McCrea Room 118). 1. Differentiate the following equations with respect to x to find dx dy 9 : (a) xey = In( 1 - x 1+x ; (b) xe y = cos y ; (c) ln(x 2 + y 2 ) = 2 arctan(y/x) . 2. Working from the definitions of cosh x = "(e" + e-ª) and sinh x = {(e" - e-a), show that (a) cosh x + sinh x = et ; (b) cosh(x + y) = cosh x cosh y + sinh x sinh y ; (c) sinh(x + y) = sinh x cosh y + sinh y cosh x ; (d) sin(ix) = i sinh x ; (e) cos(ix) = cosh x . Using these results, write down the expressions for cosh 2x and sinh 2x, in terms of cosh x and sinh x. 3(a). Solve the equation sinh x =eª+ 1 . [Hint: Solve a quadratic equation for u = er and hence derive an expression for x.] (b) Solve the simultaneous equations cosh x + cosh y = 4 ; sinh x - sinh y = 2 . 4. Differentiate the following functions with respect to x: (a) arcsinh(cos x) ; (b) arcsech(=). (c) x-2 cosh 2x ; cosh x + sinh x ; (d) cosh x - sinh x Please turn over
5. (a) Given that y = sin x 1-x2 ' show that (1- x2) dx2 ?2 y - 4x- dx dy - (1 + x2) y = 0 . Hint: Leibniz formula can be very useful here! (b) Given that y = sin(a arcsin x) , (1- x2) ?2 y dx2 dy - x- dx + a2 y = 0 . for real constant a, show that (c) Given that y = e"ka ( a cos nx + bsin nx) , for real constants a, b, n and k, show that ?2 y dy dx2 + 2k- dx + (n2 + k2)y = 0 . Hint: Again, Leibniz formula can be very useful here!