FINAL HONOURS.

DIFFERENTIAL EQUATIONS.

1. Solve (a) (9x2y — 4xy6)dx — 2(6x3 +x2y3)dy=o. (b) (3x+5v+6)dy − (7y+x+2)dx=0.

(c) the 1st order linear equation.

(d) dy-(y sin x + sin 2x)dx = o.

2. (a) Show that the complete primitive of p2+ pp(x,y)+f(x,y) =o is of the form c2+cp1(x,y) + f1(x,y)=0.

(b) Show that y2=4ax is a solution of xp2 −yp+a =o, and explain how this can be so in view of the truth of (a). Explain completely the general theory

underlying this.

3. From the p,r equation of the circle, find the differential and cartesian equations of the cycloid.

5. Express the equation (x3D3 − 2x2 D2 +4xD−4)v =o in terms of the independent variable z, where z= log x. Thence solve the equation.

6. Show how the linear equation

d2y

+pdy

+ Qy=X dx2 dx

may be reduced by a change of variable to the linear

DIFFERENTIAL CALCULUS.

FINAL HONOURS.

1. (a) Establish Leibnitz's theorem for the nth differentiation of a product of two functions.

(b) If y=r" log r, show that nc, log is the only term containing r in yn

2. Find der/dy in terms of r when yr log x.

3. (a) State the conditions under which fry)= 0 and (ry)=0 have contact.

(b) Find the conditions under wihch

cos y sin 0-0 will touch am/am+m/bm=1.

4. In the catenary y=cosh(.r/c) the perpendicular from the foot of the ordinate to the tangent is constant.

5. Find the tangential equation of ar′+by”+c=0, where z is a unit variable.

6. Given r=ft and y=pt, determine the radius of curvature in terms of t.

7. Show that at points of greatest and least curvature the circle of curvature has 3rd order contact.

8. Find the polar reciprocal of a parabola with focus as pole.

sin y=a, show that the max. value of

9. Given cos sin r+cos y is V (4-a2).