MATHEMATICS - II
UNIT – I
ORDINARY DIFFERENTIAL EQUATIONS
Part – A
1. Find the particular integral of D2+1y=sinx.
2. Solve D4-1y=0.
3. Find the particular integral of D2-2D+2y=excosx .
4. Solve the equation D2-6D+13y=0.
5. Find the particular integral of (D+1)2y=e-xcosx.
6. Solve D3-3D2-D+3y=0.
7. Find the particular integral of D2-4D+3y=e2xcos2x .
8. Solve D2+1y=e-x.
9. Find the particular integral of D2+1y=cosh2x.
10. Write the general solution for the method of variation of parameters for second order differential equation.
11. Solve (x2D2+xD)y=0.
12. Transform the equation x2y"+xy'=x into a linear differential equation with constant coefficients.
13. Reduce the equation (x2D2+xD+1)y=logx.
14. Solve x2y"-20y=0.
15. Convert the ordinary differential equation (2x+5)2y"-6(2x+5)y'+8y=0 into an equation with constant coefficients.
16. Transform (1+x)2d2ydx2+1+xdydx+y=2sin(log(1+x)) into linear equation with constant coefficients.
17. Transform (x+2)2d2ydx2-x+2dydx+y=3x+4 into differential equation with constant coefficients.
18. Form the ordinary differential equation in y by eliminating x from dxdt=3x+8y, dydt=-x-3y.
19. Eliminate y and z form the system dxdt=2y, dydt=2z,dzdt=2x.
Part – B
1. Solve 3D2+D-14y=5+13e2x
2. Solve D2-2D+1y=(ex+1)2
3. Solve D2+5D+6y=e-7xsinh3x.
4. Solve the equation D2-3D+2y=2cos2x+3+2ex.
5. Solve D2+3D+2y=sin3xcos2x.
6. Solve D2+9y=11cos3x.
7. Solve D3-D2-6Dy=x2+1.
8. Solve D2-4D+13y=e2xcos3x+x2+x+9.
9. Solve d2ydx2-2dydx+y=exxcosx.
10. Solve D2-3D+2y=e-xsin2xcosx.
11. Solve the equation D2+4D+3y=e-xsinx.
12. Solve D2-2D+1y=xsinx.
13. Solve the equation D2+1y=x sinx by method of variation of parameters.
14. Solve by method of variation of parameters d2ydx2+4y=sec2x.
15. Solve the differential equation y"+ay=secax by the method of variation of parameters.
16. Apply the method of variation of parameters to solve D2+4y=cot2x.
17. Solve D2-4D+4y=e2x by the method of variation of parameters.
18. Solve d2ydx2+4y=4tan2x using method of variation of parameters.
19. Solve the equation (x2D2-2xD+4)y=x2+2logx .
20. Solve (x2D2+3D+1)y=sin(logx)x2.
21. Solve (x2D2-3xD+4)y=x2cos(logx) .
22. Solve the differential equation (x2D2-xD+4)y=x2sin(logx).
23. Solve x2d2ydx2+xdydx-9y=10+5x2.
24. Solve x2d2ydx2+xdydx+y=logxsin(logx).
25. Solve (2x+5)2y"-6(2x+5)y'+8y=6x
26. Solve (1+4x)2y"+(1+4x)y'+4y=8(1+4x)2
27. Solve: dxdt+2x+3y=2e2t, dydt+3x+2y=0.
28. Solve: dxdt+2y=-sint, dydt-2x=cost given x = 1 and y = 0 at t = 0.
29. Solve the simultaneous equation dxdt+2x-3y=5t , dydt-3x+2y=2e2t.
30. Solve the simultaneous differential equation dxdt+2y=sin2t , dydt-2x=cos2t.
31. Solve dxdt+2y=5et , dydt-2x=5et.
UNIT – II
VECTOR CALCULUS
Part – A
1. Prove that ∇rn=nrn-2r.
2. Prove that ∇r2=2r.
3. Find grad1r and grad logr.
4. Find the unit normal vector to the surface x2+xy+z2=4 at (1,-1,2).
5. Find a unit normal vector to the surface x2+y2-z=10 at (1,1,1).
6. Find the maximum directional derivative of ϕ=x2yz+4xz2 at the point P(1,-2,-1).
7. If the directional derivative of the function φ=xyz at (1,1,1) in the direction of αi+j+k is 3, find α.
8. If r=xi+yj+zk then find div r and curl r.
9. Prove that the curl of costant vector is zero.
10. If φ=3x2y-y3z2, find grad φ at (1,-1,2).
11. Find λ such that F=(3x-2y+z)i+(4x+λy-z)j+(x-y+2z)k is solenoidal.
12. Show that F=y2-z2+3yz-2xi+3xz+2xyj+(3xy-2xz+2z)k is solenoidal.
13. If v=x+3yi+y-2zj+(x+λz)k is solenoidal, find the value of λ.
14. Find div curlF where F=x2yi+xzj+2yzk.
15. Evaluate C(5y2dx-2x2dy) along the parabola y=x2 from (0,0) to (2,4).
16. Evaluate (xdy-ydx) around the circle x2+y2=1.
17. State Gauss divergence theorem.
18. State Stoke’s theorem.
19. State Green’s theorem.
20. Using Stoke’s theorem, prove that curl grad φ=0.
21. If S is any closed surface enclosing the volume V and r is the position vector of a point, prove that Sr.nds=3V.
22. If F is irrotational and C is a closed curve then find the value of CF.dr.
Part – B
1. Find a and b such that the surfaces ax2-byz=a+2x and 4x2y+z3=4 cut orthogonally at (1,-1,2).
2. Show that F=y2+2xz2i+2xy-zj+(2x2z-y+2z)k is irrotational and hence find its scalar potential.
3. Find the scalar potential φ if ∇φ=y2+2xz2i+2xy-zj+(2x2z-y+2z)k.
4. Find the scalar scalar potential of F=6xy+z3i+3x2-zj+(3xz2-y)k.
5. Find the total work done in a moving particle in a force field given by F=(2x-y+z)i+(x+y-z)j+(3x-2y-5z)k along a circle C in XY plane x2+y2=9,z=0.
6. If F=x2i+xyj , evaluate F.dr from (0,0) to (1,1) along the line y = x.
7. Evaluate CF.dr where F=3x2i+(2xz-y)j+zk and C is the straight line from A(0,0,0) to B(2,1,3).
8. Given the vector field F=xzi+yzj+z2k, evaluate CF.dr from the point (0,0,0) to (1,1,1) where C is the curve x=t,y=t2, z=t3.
9. Evaluate SF.n ds where F=zi+xj-y2zk and S is the part of the surface of the cylinder x2+y2=1 included in the first octant between the planes z = 0 and z = 2.
10. Evaluate SF.n ds where F=18zi-12j+3yk 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 F=4xzi-y2j+yzk over the cube bounded by x = 0, y = 0, z = 0, x = 1, y = 1, z = 1.
12. Verify Gauss divergence theorem for F=(x2-yz)i+(y2-zx)j+(z2-xy)k taken over the rectangular parallelepiped 0≤x≤a,0≤y≤b,0≤z≤c.
13. Verify Gauss divergence theorem for F=x2i+y2j+z2k, 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 F=4xi-2y2j+z2k taken over the region bounded by x2+y2=4,z=0 and z = 3
15. Verify Green’s theorem in a plane for C[3x2-8y2dx+4y-6xydy] , where C is the boundary of the region defined by x = 0, y = 0 and x + y = 1.
16. Using Green’s theorem, evaluate C[2x2-y2dx+x2+y2dy]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 Cxy+y2dx+x2dy where C is the closed curve of the region bounded by y = x and y = x2.
18. Verify Green’s theorem for Cx2-y2dx+2xydy, 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 CF.dr , where F=y2i+x2j-(x+z)k 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 F=x2i+xyk 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 F=x2-y2i+2xyj in the rectangular region XOY plane bounded by the lines x = 0 , x = a , y = 0 , y = b.
22. Prove ∇X(F±G)=∇XF±∇XG or curlF±G=curl F±curl G.
23. Show that ∇XφF=∇φXF+φ(∇XF) or curlφF=gradφ X F+φ curl F.
24. Prove ∇.FXG=G.∇XF-F.(∇XG) or divFXG=G.curl F-F.curl G.
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) = z or w = x – iy is analytic or not.
3. Check whether the function 1z is analytic or not.
4. Is f(z) = z3 analytic? Justify.
5. Show that the function fz=x3-3xy2+i(3x2y-y3) satisfies Cauchy - Riemann equations.
6. Show that an analytic function with constant real part is zero.
7. Prove that fz=zn is analytic
8. Show that tan-1yx is harmonic.
9. Verify if the function e-2xcos2y can be the real or imaginary part of an analytic function.
10. Prove that the function u=3x2y+2x2-y3-2y2 is a harmonic function.
11. Show that the real part of an analytic function u satisfies the Laplace equation ∇2u=0.
12. If w = ez , find dwdz 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 w=(z-α)(z-β).
15. Prove that a bilinear transformation has atmost two fixed points,
16. Find the fixed points of the bilinear transformation w=z-1z+1.
17. Find the invariant point of the transformation w=1z-2i.
18. Find the image of the circle z=3 under the transformation w = 5z.
19. Find the image of the circle z=2 by the transformation w = z + 3 + 2i.
20. Find the image of the circle x2+y2=4 under the transformation w = 3z.
Part - B
1. Show that the function u=x3-3xy2+3x2-3y2+1 is harmonic and find its analytic function.
2. Show that u=ex(xcos y-ysiny) is harmonic and hence find the analytic function.
3. If u=2sin2xe2y+e-2y-2cos2x, find w = f(z) such that f(z) is analytic.
4. Find the analytic function u=(x-y)(x2+4xy+y2). Also find the conjugate harmonic function v.
5. Show that the function u=12log(x2+y2)is harmonic and determine its conjugate. Also find f(z).
6. Prove that v=x2-y2+e-2xcos2y is harmonic and find its harmonic conjugate.
7. Find the analytic function f(z) = u + iv where v=2sinxsinhycos2x+cosh2y.
8. Find the analytic function f(z) = u + iv, given that u+v=sin2xcosh2y-cos2x.
9. Find the analytic function f(z) = u + iv, if u+v=xx2+y2 and f(1) = 1.
10. If f(z) is an analytic function and u-v=ex(cosy-siny), find f(z) interms of z.
11. If f(z) is an analytic function of z, prove that ∂2∂x2+∂2∂y2logf(z)=0.
12. If f(z) is an analytic function of z, prove that ∇2f(z)2=4f'(z)2.
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) 14<y<12 and (ii) 0<y<12 under the transformation w=1z.
16. Find the image of the circle z-1=1 in the complex plane under the mapping w=1z.
17. Find the image of the circle z-2i=2 under the transformation w=1z.
18. Find the image of the half plane x > c, c > 0 under w=1z. Sketch graphically.
19. Find the bilinear transformation which maps the points z=0,1,∞ into w=i,1,-i.
20. Find the bilinear transformation which maps the points z=1,i,-1 into w=i,0,-1.
21. Find the bilinear transformation which maps the points z1=-1,z2=0,z3=1 into the points w1=0,w2=i, w3=3i.
22. Find the bilinear transformation that transforms 1,i,and -1 of 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.