Solve all Q31 explaining detailly each step II. b) Application:28. Find the area of the finite region in the first quadrant bounded by the curves y = e°, y=e*,x = In2.d²ydy+ 13y = 03x29. a) Given that y = e*sin2x, show that:dx²dx%3Db) Solve the differential equation (x- 4)-- + 2xy = 0, (x ±2). given that y = 1/5 when x =dx3.30. i) Find in the form y = f(x), the general solution of the differential equation:dyx²-1(1+x*)–-2x(1-y)= 0. Given that y = 0 when x = 1, show that f(x)31. Find the area of the finite region bounded by the curves y = x(10 - x) and y = x (3x+ 2).|dxx²+1dy32. Show that the solution of the differential equation (x + x – 2)-+ 2x + 1 =0 for which y=di1- and x= 2 is y= Sketch the integral curve corresponding to this solution showing4(x+1)(x+2)clearly all the vertical and horizontal asymptotes and turning points.áy33. Solve in the form y = f(x), the differential equation: x= v (2x + 1). Hence on a separatediagram, skeich aiso the graph of y = f(x)|.34. Find the area of the finite region enclosed by the curve xy = 1 and line x+y = 4, leavingyour answer in terms of naturai logarithms.35. Find the total area of the finite regions bounded by the crve y = x(2- X), the x-axis and theordinate at x =3D3.36. Solve the differential equation (x-2)(x+3)-- (2x+1)= 0 given that y= In6 when x =dy3637. Find the general solution of the differential equation: x = Inx + y Inx, expressing yourdydxanswer in the form y = f(x).38. The table given values of a continuous variable y corresponding to the given values of x.1372765119a. Use the trapezium rule to find an estimate for vdx.b. A relation of the form y = ax´+bx is known to exist between x and y. By plotting-againstX. estimate the values of the constant a and b to the nearest interger. Hence, obtain another7estimate for f ydx by direct intergration.(39. Find the area of the region enclosed by the curve with equation: y = x(2 – x) and theliney = x.602.

Question

Accepted Answer

Q 31
Consider the curves,
y=x10-x   and    y=x3x+2
To calculate the intersecting points between the…