MATHEMATICS - II
UNIT – I
ORDINARY DIFFERENTIAL EQUATIONS
Part – A
1. Find the particular integral of .
2. Solve .
3. Find the particular integral of .
4. Solve the equation .
5. Find the particular integral of .
6. Solve .
7. Find the particular integral of .
8. Solve.
9. Find the particular integral of .
10. Write the general solution for the method of variation of parameters for second order differential equation.
11. Solve .
12. Transform the equation into a linear differential equation with constant coefficients.
13. Reduce the equation .
14.Solve .
15. Convert the ordinary differential equation into an equation with constant coefficients.
16. Transform into linear equation with constant coefficients.
17. Transform into differential equation with constant coefficients.
18. Form the ordinary differential equation in y by eliminating x from .
19. Eliminate y and z form the system .
Part – B
1. Solve
2. Solve
3. Solve .
4. Solve the equation .
5. Solve .
6. Solve .
7. Solve .
8. Solve .
9. Solve .
10. Solve .
11. Solve the equation .
12. Solve .
13. Solve the equation by method of variation of parameters.
14. Solve by method of variation of parameters .
15. Solve the differential equation by the method of variation of parameters.
16. Apply the method of variation of parameters to solve .
17. Solve by the method of variation of parameters.
18. Solve using method of variation of parameters.
19. Solve the equation .
20. Solve .
21. Solve .
22. Solve the differential equation .
23. Solve .
24. Solve .
25. Solve
26. Solve
27. Solve: .
28. Solve: given x = 1 and y = 0 at t = 0.
29. Solve the simultaneous equation .
30. Solve the simultaneous differential equation .
31. Solve .
UNIT – II
VECTOR CALCULUS
Part – A
1. Prove that .
2. Prove that .
3. Find and .
4. Find the unit normal vector to the surface at (1,-1,2).
5. Find a unit normal vector to the surface at (1,1,1).
6. Find the maximum directional derivative of at the point P(1,-2,-1).
7. If the directional derivative of the function at(1,1,1) in the direction of is , find .
8. If then find div and curl .
9. Prove that the curl of costant vector is zero.
10. If , find at (1,-1,2).
11. Find such that is solenoidal.
12. Show that is solenoidal.
13. If is solenoidal, find the value of .
14. Find where .
15. Evaluate along the parabola from (0,0) to (2,4).
16. Evaluate around the circle .
17. State Gauss divergence theorem.
18. State Stoke’s theorem.
19. State Green’s theorem.
20. Using Stoke’s theorem, prove that .
21. If S is any closed surface enclosing the volume V and is the position vector of a point, prove that .
22. If is irrotational and C is a closed curve then find the value of .
Part – B
1. Find a and b such that the surfaces and cut orthogonally at (1,-1,2).
2. Show that is irrotational and hence find its scalar potential.
3. Find the scalar potential if .
4. Find the scalar scalar potential of .
5. Find the total work done in a moving particle in a force field given by along a circle C in XY plane .
6. If , evaluate from (0,0) to (1,1) along the line y = x.
7. Evaluate where and C is the straight line from A(0,0,0) to B(2,1,3).
8. Given the vector field , evaluate from the point (0,0,0) to (1,1,1) where C is the curve .
9. Evaluate where and S is the part of the surface of the cylinder included in the first octant between the planes z = 0 and z = 2.
10. Evaluate where and S is the part of the surface of the plane 2x + 3y + 6z = 12 which is in the first octant.
11. Verify Gauss divergence theorem for over the cube bounded by x = 0, y = 0, z = 0, x = 1, y = 1, z = 1.
12. Verify Gauss divergence theorem for taken over the rectangular parallelepiped .
13. Verify Gauss divergence theorem for , where S is the surface of the cuboid form by the planes x = 0, x = 1 , y = 0 , y = 2 , z = 0, z = 3.
14.Verify Gauss divergence theorem for taken over the region bounded by and z = 3
15. Verify Green’s theorem in a plane for , where C is the boundary of the region defined by x = 0, y = 0 and x + y = 1.
16. Using Green’s theorem, evaluate where C is the boundary of the square enclosed by the lines x = 0 , y = 0, x = 2, and y = 3.
17. Verify Green’s theorem in the plane for where C is the closed curve of the region bounded by y = x and y = x2.
18. Verify Green’s theorem for , where C is the boundary of the rectangle in the XOY – plane bounded by the lines x = 0, x = a , y = 0 , y = b.
19. Using Stoke’s theorem, evaluate , where and C is the boundary of the triangle with vertices (0,0,0), (1,0,0),(1,1,0).
20. Verify Stoke’s theorem for the function integrated around the square in the z = 0 plane whose sides are along the lines x = 0, x = a , y = 0, y = a.
21. Verify Stoke’s theorem for in the rectangular region XOY plane bounded by the lines x = 0 , x = a , y = 0 , y = b.
22. Prove or .
23. Show that or .
24. Prove or .
UNIT – III
ANALYTIC FUNCTIONS
Part - A
1. State the basic difference between the limit of a function of a real variable and that of a complex variable.
2. Verify whether f(z) = or w = x – iy is analytic or not.
3. Check whether the function is analytic or not.
4. Is f(z) = z3 analytic? Justify.
5. Show that the function satisfies Cauchy - Riemann equations.
6. Show that an analytic function with constant real part is zero.
7. Prove that is analytic
8. Show that is harmonic.
9. Verify if the function can be the real or imaginary part of an analytic function.
10. Prove that the function is a harmonic function.
11. Show that the real part of an analytic function u satisfies the Laplace equation .
12. If w = ez , find using complex vaiable.
13. Define critical point of a conformal mapping w = f(z). Also find the critical points of w = z2.
14. Find the critical points of the transformation .
15. Prove that a bilinear transformation has atmost two fixed points,
16. Find the fixed points of the bilinear transformation .
17. Find the invariant point of the transformation .
18. Find the image of the circle under the transformation w = 5z.
19. Find the image of the circle by the transformation w = z + 3 + 2i.
20. Find the image of the circle under the transformation w = 3z.
Part - B
1. Show that the function is harmonic and find its analytic function.
2. Show that is harmonic and hence find the analytic function.
3. If, find w = f(z) such that f(z) is analytic.
4. Find the analytic function . Also find the conjugate harmonic function v.
5. Show that the function is harmonic and determine its conjugate. Also find f(z).
6. Prove that is harmonic and find its harmonic conjugate.
7. Find the analytic function f(z) = u + iv where .
8. Find the analytic function f(z) = u + iv, given that .
9. Find the analytic function f(z) = u + iv, if and f(1) = 1.
10. If f(z) is an analytic function and , find f(z) interms of z.
11. If f(z) is an analytic function of z, prove that .
12. If f(z) is an analytic function of z, prove that .
13. When the function f(z) = u + iv is analytic, prove that the curves u = constant and v = constant are orthogonal.
14. Find the image of the region y > 1 under the transformation w = (1 – i ) z.
15. Find the image of the infinite strips (i) and (ii) under the transformation .
16. Find the image of the circle in the complex plane under the mapping .
17. Find the image of the circle under the transformation.
18. Find the image of the half plane x > c, c > 0 under . Sketch graphically.
19. Find the bilinear transformation which maps the points into .
20. Find the bilinear transformation which maps the points into .
21. Find the bilinear transformation which maps the points into the points .
22. Find the bilinear transformation that transforms1,i,and -1of the z-plane onto 0,1, of the w-plane. Also show that the transformation maps interior of the unit circle onto upper half of the w-plane.