FINAL HONOUR MATHEMATICS INTEGRAL CALCULUS. Any seven questions. 1. Show that the integral of a rational function consists of rational terms, and transcendental terms which either logarithms or inverse tangents. are 2. If f(x) is finite and continuous on the finite interval (a, b), show that 3. (a) If f(a) is infinite, state the conditions under which Sf(x) dx α shall be finite and prove that these conditions are sufficient. (b) Apply (a) to test the convergence of the following definite integrals: (4) x-(1-x)-1 dx. 4. Evaluate (1) fx2 sin−1x dx (2) S (3) 1, 1 0, 0 2 2 x2 e-a x cos bx dx, (x2 +y) dx+(x+y) dy, along y3 = x2. 5. Evaluate the following integrals, giving the geometric interpretation of the substitution used in each case, as a power series in k2, giving an expression for the general term. (b) Define the elliptic functions am u, sn u, cn u, and dn u, and obtain their derivatives with respect to u. 7. Obtain a Fourier series which will represent the function f(x)=x,0<x<π/2; f(x)=−x, −π/2<x<0. Indicate how a different series might be obtained to represent the same function. 8. Find the total area bounded by the curves y2=ax and (2a—x) y2=x3. г(a) = ƒ ° xa¬1e ̄* dx. Make a table of г(x) for every integer and half integer between 0 and 5, inclusive, and plot the curve y=r(x). 10. Find the volume of two equal cylinders of radius a which intersect centrally at right angles. FINAL HONOUR MATHEMATICS DIFFERENTIAL EQUATIONS. Any seven questions. 1. (a) Solve the equation f(x, y)dx+p(x, y)dy=0 where ƒ and are homogeneous functions of degree n. (b) Reduce (2x—y—1)dx—(x+4y—5)dy—0 to the form in (a), and solve the equation. 2. (a) Find a system of integrating factors for ray(mydx+nxdy)=0. (b) Solve (y3-2x2y) dx+(2xy2—x3)dy=0. Illus 3. Explain the use of the p and c discriminants in finding. the geometrical meaning of a differential equation. trate with the equation pr=(x-a)2. 4. Solve (a) (1-x2) dy -xy=axy2. dx (b) y=p2—2p3. 5. Find a curve such that the part of the tangent intercepted between the coordinate axes is constant. 6. Give the geometric interpretation of the differential equation of the second order, and show from it that the complete primitive must have two arbitrary constants. Illustrate with the equation |