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1476375257175SNS COLLEGE OF ENGINEERING, CBE – 107QUSTION BANK MATHEMATICS – II UNIT-1 (ORDINARY DIFFERENTIAL EQUATIONS)PART-ADefine linear differential equations Solve d2ydx2-.6 dydx +13y = 0Solve (D2 + 1) y =0 given y(0) =0 , y’(0)=1Solve (4D2-4D +1)y = 4Solve (D2_1)y = xSolve the equation x2 y11-xy1+y= 0Solve (D+2)2y = e-2xsinxFind the particular integral of (D2+4)y = cos2xFind the particular integral of (D2+1)y = sinxFind the particular integral of (D2+1)y = xexFind the particular integral of (D2_4)y = cosh2xFind the particular integral of (D-1)2y = exsinxFind the particular integral of (D2-2D+5)y = ex cos2x Find the particular integral of (D2-2D+1)y = ex ( 3x2-2) Find the particular integral of (D2-4D+4)y = 2x 16) Solve (D2+2D +1) y =π17) Solve (D2-3D -4) y = e 3x +e-x18) (D2+1) y = sin2x19) Transform the equation x2 y11+xy1= x into a linear differential equation with constant Co-efficient.20) Transform the equation (2x+3)2 d2ydx2 - 2(2x+3) dydx - 12y =6x into a linear differential Equation with constant co-efficients21) Find the Wronskian of y1, y2 of y11 -2y 1 +y =ex logxPart-B Find the particular integral of (D2+4)y = x2 cos2xSolve (D2+3D+2)y = sin3x cos2xSolve (D2+5D+6)y = e –7x sinh3xSolve (D3 - 3D2 + 4D-2)y = sinh2x (D2+9)y = 11 cos3xSolve (D2-6D+13)y =8 e 3x sin4xSolve (D2+4D+4)y = e –2x/ x2 Solve (D2-2D+1)y = e x xcosxSolve the equation (D2+a2)y = secax by the method of variation of parametersSolve d2ydx2+ 4y= 4tan2x by the method of variation of parametersSolve d2ydx2+ y= cosecx by the method of variation of parametersSolve (D2+1)y = xsinx by the method of variation of parametersSolve (D2-2D+1)y = e x logx by the method of variation of parametersSolve(1+x2)2 d2ydx2 +(1+x) dydx+y = 2sin(log(1+x))Solve (x2 d2ydx2 -2x dydx -4)y = x2+2logxSolve(x2D2+ xD+1)y= logxsin(logx)Solve(x2D2+ xD+4)y=cos(logx)+ xsin(logx)Solve (D2+1XD)y =12logxx2Solve(x2D2+ 3D+1)y= sin(logx)/x2Solve((3x+2)2d2ydx2 +3(3x+2) dydx -36)y =3x2+4x+1Solve(x+1)3d2ydx2 +3(x+1)2 dydx +(x+1)y =6log(x+1)Solve the system of equations dxdt+y =et ;x- dydt = tSolve the system of equations dxdt+y =sint ; dydt+x=cost given that x=2,y=0 when t=0 Solve the system of equations dxdt+2x+3y =2e2t ; dydt+3x+2y =0 Solve the system of equations dxdt-y =t ; dydt+x =t2 Solve the system of equationsdxdt-dydt +2y=cos2t ; dxdt+dydt -2x =sin2tSolve the simultaneous differential equations dxdt +2y=sin2t ; dydt -2x =cos2tUNIT II – (VECTOR CALCULUS)Part – ADefine vector Differential operator(?)Define gradient of the scalar function φ.If f and g are two scalar point function then ?(fg) = f?g + g?f.If φ = log(x2+y2+z2) find ?φ.Prove that ?fr=f'rrr ,r = xi +yj +zk .Find the directional derivative of φ=xy+yz+zx in the direction vector i +2j +2k at (1, 2 ,0)Find the directional derivative of φ=3x2+2y-3z at (1 , 1 ,1) in the direction of 2i +2j-k .Find the unit vector normal to the surface x2-y2+z=2 at the point (1 , -1 ,2)Find the unit vector normal to the surface x2y+2xz2=8 at the point(1 , 0, 2)Prove that ?×(F ±G )=?×F ± ?×G .Prove that ??F ×G =G ??×F –F ??×G .Find ??F and ?×F of the vector point function F = xz3 i-2x2y z j +2yz4k at the point (1 ,-1 ,1).Prove that curl ( grad φ) = 0.Prove that div (grad φ) = ?2φ.Show that the vector F = 3y4z2 i+4x3z2 j-3x2y2k is solenoidal and 2xyi +x2+2yzj +(y2+1)k is irrotational.Find the value of a , b , c so that the vector F =x+2y+az i +bx-3y-zj +(4x+cy+2z)k is irrotational.If F =x2i +xy j , evaluate F ?dr from (0 ,0) to (1 ,1) along the line y = x.Find the value of ‘a’ given two vectors 2i -3j +5k and 3i +aj -2k are perpendicular.If r = xi +yj +zk . S is the upper half surface of the sphere x2+y2+z2=a2,then find Sr ?n ds If v is the volume of the region enclosed by the cube 0 < x ,y ,z <1 and F =x2i +y2j +z2k , then V??F dV isPart – BIf r = xi +yj +zk prove that (i) ?r=r r , (ii) ?rn=nrn-2 r ,where r=|r |. Find the angle of intersection at the point (2 ,-1,2) of the surfaces x2+y2+z2=9 and z=x2+y2-z-3.Find ‘a’ and ‘b’ such that the surfaces ax2-byz=a+2x and 4x2y+z3=4 cut orthogonally at (1 ,-1,2).If ?φ=2xyzi +x2zj +x2yk , find the scalar potential φ.Evaluate CF ?dr where F = 3x2 i+2xz-y j+zk and C is the straight line from A(0 ,0 ,0) to B(2 , 1, 3).Given the vector field F = xz i +yz j-z2k , evaluate CF ?dr from the point (0,0,0) to (1,1,1) where C is the curve (i) x = t , y = t2, z = t3, (ii) the straight path from (0,0,0) to (1,1,1).Find the total work done in moving a particle in a force field given by F =2x-y+z i +x+y-zj +(3x-2y-5z)k along a circle C in the XY plane x2+y2=9, z=0.Find the work done by the force F =2xy+z3 i +x2j +3xz2k when it moves a particle from (1,-2,1) to (3,1,4) along any path.Evaluate where and S is that part of the surface of the sphere x2+y2+z2 = 1 which lies in the first octant.Evaluate where as S is the part of the plane 2x + 3y + 6z = 12 which is in the first octant.Evaluate where where S is the region bounded by 2x + y + 2z = 6 in the first octant.If , then evaluate (i) (ii) ,where V is the region bounded by x = 0 , y = 0 , z = 0 and 2x + 2y + z = 4.Verify the Gauss divergence theorem for F =4xz i -y2j +yzk over the cube bounded by x = 0 , x = 1, y = 0, y = 1, z = 0, z = 1.Verify the 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 .Evaluate where and S is the surface of the cube bounded by x = 0 ,x = 1, y = 0, y = 1, z = 0, z = 1.Use divergence theorem to evaluate F =4x i -2y2j +z2k and S is the surface bounding the region x2 + y2 = 4 z = 0 and z = 3.Verify Green’s theorem in a plane for the integral ,taken around the circle x2 + y2 = 1.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 and y = a.Verify Green’s theorem for , where C is the closed curve of the region bounded by y = x and y = x2 .Using Green’s theorem, evaluate ,where C is the triangle bounded by y=0,x=π2 , y=2xπ.By applying Green’s theorem prove that the area bounded by a simple closed curve C is = and hence find the area of the ellipse.Verify Stoke’s theorem for a vector defined by F = (x2-y2) i+2xy j in the rectangular region in the XOY plane bounded by the lines x = 0, x = a, y = 0 and y = b.Verify Stoke’s theorem for a vector defined by F = y i+z j+xk ,where S is the upper half of the surface of the sphere x2 + y2 + z2 = 1 and C is its boundary.Evaluate the integral where C is the boundary of the triangle with vertices (2,0,0), (0,3,0) and (0,0,6) using Stoke’s theorem.Evaluate by the Stoke’s theorem where C is the square in the XY plane with vertices (1,0), (-1,0), (0,1) and (0,-1).Prove that where S is the surface enclosing a circuit C.UNIT 3 - (ANALYTIC FUNCTIONS – COMPLEX VARIABLES)Part AShow that xx2+y2 is harmonic.Is f(z) = z3 analytic?Find the invariant point of the transformation w = 1z-2iShow that xy2 cannot be the real part of an analytic function.Find the image of x2+y2 = 4 under the transformation w=3z.f(z) = u + iv is such that u and v are harmonic is f(z) analytic always? Justify.State the Cauchy-Riemann equations in polar coordinates satisfied by an analytic function.Find the invariant points of the transformation w=2z+6z+7 .Find the analytic region of f(z) = (x-y)2+2i (x+y).Find the critical points of the transformation w2= (z-α)(z-β).Define conformal mapping.For what values of a,b and c the function f(z) = x – 2ay + i(bx-cy) is analytic?If u+iv is analytic, show that v-iu is also analytic.Find the image of the circle |z| = 2 under the transformation w = 3z.Define bilinear transformation.Define analytic function of a complex variable.Give an example such that u & v are harmonic but u+iv is not analytic.Find ‘a’ so that u(x,y) = ax2-y2+xy is harmonic.State the orthogonal property of an analytic function.Under the transformation w = iz + I show that the half plane x>0 maps into the half plane w>1.Find the points in the z plane at which the mapping w = z + z-1, (z≠0) fails to be conformed.Prove that tan-1yx is harmonic.Show that the function f(z) = zz is not analytic at z=0.f(z) = r2 (cos2θ+i sinpθ) is analytic if the value of p is ….?? b) 0 c) 2 d) 1Define Mobius transformation.If u ≠ iv = 1z; then prove that the families of curves u = c1 and v = c2 ( c1 , c2 being constants) cut orthogonally.Define Isogonal transformation.Verify whether w = sinx coshy+ i cosx sinhy is analytic or not.Find the bilinear transformation which maps the points z = -2, 0, 2 into the points w = 0, i, -i respectively.If f(z) is a regular function of z, prove that (?2?x2 + ?2?y2) |f(z)|2 = 4 |f ’(z)|2Part BFind the analytic function whose real part is sin2xcosh2y- cos2x .Find the image of the infinite steps? < y < ? 0 < y < ? under the transformation w = 1zFind the bilinear transformation which maps -1, 0, 1 of the z-plane onto -1, -i, 1 of the w-plane. Show that under this transformation the upper half of the z-plane maps onto the interior of the unit circle |w|=1.Find the analytic function f(z) = u + iv, where v = 2 sinx sinhycos2x+ cosh2y .Find the bilinear transformation which maps the points z = 0, 1, ∞ into w=i,-1,-i.Prove that x2 – y2 + e-2x cos2y is harmonic and find its harmonic conjugate.If φ and Ψ are functions of x and y satisfying Laplace equation namely ?2φ?x2 + ?2φ?y2 = 0; ?2Ψ?x2+?2Ψ?y2 = 0 and u = ?φ?y- ?Ψ?x ; v = ?φ?x+ ?Ψ?y . show that u +iv is analytic.Show that the function f(z) = |xy| is not regular at the origin, through C-R equations are satisfied at origin.Determine the region D’ of the w-plane into which the triangular region D enclosed by the lines x=0, y=0, x+y=1 is transformed under the transformation w = 2z.Find the analytic function f(z) = u +iv if u + v = xx2+y2 and f(1)=1.Show that the mapping w = i+zi-z , the image of the circle x2+y2 < 1, is the entire half of the w-plane to the right of the imaginary axis.Show that an analytic function with constant modulus is also constant.Find analytic function f(z)=u(r,θ)+iv(r,θ) such that v(r,θ)=r2cos2θ-rcosθ+2. UNIT – IV(COMPLEX INTEGRATION)PART -AEvaluate 01+ix-y+ix2dz along the line from z = 0 to 1+i.Evaluate csinz dz along the line z=0 to z=i.Prove that c(z-a)n dz=0, [n=-1] where c is the circle. |Z-a|=rEvaluate 02+i(Z)2dz along the line y=x2State Cauchy’s integral theorem.Evaluate cdz2z-3 where c is the circle |Z|=1.Evaluate cdzzez where c is the circle |Z|=1.Evaluate ce1z where c is the circle |Z|=1.Define Taylor’s series.Define Laurent’s seriesDefine Singularity.Find the residue of f(z)= 1z2ezFind the residue of f(z) = z+1z2(z-2) at each of the poles.State Cauchy’s Residue theorem.State Jordan’s LemmaPART _ BEvaluate cz2 dz where the ends of c are A(1,1) and B(2,4) given that (i)C is a curve y=x2(ii)C is the line y=3x-2Evaluate , using cauchy’s integral formula 12πi cz2+5z-3 dz on the circles (i) |z|=4 and |z|=1Show that c(Z+1) dz = 0 where C is the boundary of the square whose vertices are at the point Z=0,Z=1,Z=1+I,Z=i.Using Cauchy’s integral formula find the value of c(z+4)dzz2+2z+5 where c is the circle |z+1-i|=2Evaluate ce2zz-1z-2 dz where c is the circle |z|=3Evaluate csinπz2+cosπz2z-1z-2 dz where c is the circle |z|=3Evaluate c3z2+zz2-1 dz where c is the circle |z-1|=1Evaluate c(z+1)dzz2+2z+4 where c is the circle |Z+1+i|=2Evaluate csin6z dz(z-π6)3 where c is the circle |Z|=1Evaluatectanz/2 dz(z-a)2 ,-2<a<2 where ‘C is the boundary of the square whose sides are x= ±2 and y=±2.Evaluate c1+zz3-2z2 dz where c is the unit circle |z|=1Expand cos z as a Taylors series about the points (i)Z=0 (ii) z= π/4Expand f(z) = z2-1z+3z+2 in a Laurent’s series if (i) |z|>3 (ii) |z|<3Expand 1z2-3z+2 when 1<|z|<2 by Lauren’s Series.Obtain the Laurent’s expansion for z-2(z+2)z+1z+4 which are valid (i) 1< |z|<4 (ii) |z|>4If 0<|z-1|<2, then express f(z)= zz-1z-3 in a series of positive and negative powers of z-1.Find the residue of f(z) = z2(z-1)2z+2 at each of the poles.Find the residue of f(z) = 1(z2+1)2 about each singularity.Evaluate c(2z-1)dzzz+1(z-3) where c is the circle |z|=2Evaluate c(z2-2z)dzz+12(z2+3) where c is the circle |z|=3 using residue theoremEvaluate czsecz dz(1-z2) where c is the ellipse 4x2+9y2 = 9.Evaluate 02πdθ13+5 sinθShow that 02πdθa+b cosθ= 2πa2-b2 , a>b>002πsin2θa+bcosθdθ= 2πb2[ a-a2-b2] where 0<b< aEvaluate 02πdθ1-2asinθ+a2 , 0<a<1Evaluate 0π1+2 cosθ5+4 cosθdθProve that 0∞dx(x2+1)2= π4Evaluate -∞ ∞ x2(x2+a2)(x2+b2)dx , a>0, b>0Evaluate -∞ ∞ x2-x+2x4+10x2+9dx .Evaluate 0∞cosax(x2+1)dx, a>0 UNIT-V (LAPLACE TRANSFORM)PART-ADefine Laplace transformsFind the Laplace transform of Find the inverse Laplace transform of Find Find the Laplace transform of unit step function.State the conditions under which Laplace transform of f(t) exists.State the first shifting theorem on Laplace transforms.State the second shifting theorem on Laplace transforms.Verify initial value theorem for.Find the Laplace transform of Find the Laplace transform of Find Find Find inverse Laplace transform of Find inverse Laplace transform of If ,find Verify the finial value theorem for Find the Laplace Transform of Prove that Find Find FindFindFind Find the Laplace Transform of Define change of scale propertyFind Find Find Find the laplace transform of Define convolution theorem.Define convolution of two functions.Define initial value theorem.Define finial value theoremFind PART-BUsing convolution theorem find the inverse Laplace transform of Find using convolution theorem.Find using Convolution theoremUsing Convolution theorem, find Using Convolution theorem, find the inverse Laplace transform of Find the Laplace transform Find the Laplace transform Find the Laplace transform of square wave function defined by Find the Laplace transform of the following triangular wave function given by Find the Laplace transform of the Half wave rectifier function Find the Laplace transform of square wave function given by Verify the initial and finial value theorem for the function Verify the initial and finial value theorem for the function Solve the differential equation by using Laplace transform methodSolve the differential equation by using Laplace transform method using Laplace transform method solve the differential equation Solve using Laplace transform.Solve the differential equation using Laplace transform method.Solve using Laplace transform.Solve the differential equation using Laplace transform method.Solve using Laplace transform.Find the Laplace transform of Evaluate using Laplace transform.Find Solve the equation Find the Laplace transform of Find Find the Laplace transform of 1057275247650SNS COLLEGE OF ENGINEERING, CBE – 107INTERNAL ASSESSMENT – I(COMMON FOR ALL BRANCHES) MATHEMATICS – IIPART – ADefine linear differential equations Solve (4D2-4D +1)y = 4Solve (D2-3D -4) y = e 3x +e-xTransform the equation (2x+3)2 d2ydx2 - 2(2x+3) dydx - 12y =6x into a linear differential Equation with constant co-efficientFind the particular integral of (D2_4)y = cosh2xDefine gradient of the scalar functionφ.Find the value of a , b , c so that the vector F =x+2y+az i +bx-3y-zj +(4x+cy+2z)k is irrotational.If φ = log (x2+y2+z2) find?φ.Find the unit vector normal to the surface x2-y2+z=2 at the point (1 , -1 ,2)Prove that curl (gradφ) = 0.PART – B(a)(i) Solve the system of equations dxdt+2x+3y =2e2t ; dydt+3x+2y =0 (10) (ii) (D2+9)y = 11 cos3x(6)(OR) (b)(i) Solve ((3x+2)2d2ydx2 +3(3x+2) dydx -36)y =3x2+4x+1(10) (ii) Solve (D2-3D -4) y = e 3x +e-x(6)(a)(i)Solve (D2+3D+2)y = sin3x cos2x(8) (ii)Solve(x2D2+ xD+1) y= logx sin (logx)(8) (OR) (b)(i) Solve (4D2-4D +1) y = 4 (ii) Solve (D2+1) y = sin2x(a) Solve the simultaneous differential equations dxdt +2y=sin2t ; dydt -2x =cos2t (16) (OR) (b) 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 and y = a.(a)(i) If r = xi +yj +zk prove that (i) ?r=r r , (ii) ?rn=nrn-2 r ,where r=|r |. (ii) Prove that ??F ×G =G ??×F –F ??×G (OR) (b) Verify the 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 . 15. (a) Verify Stoke’s theorem for a vector defined by F = (x2-y2) i+2xy j in the rectangular region in the XOY plane bounded by the lines x = 0, x = a, y = 0 and y = b. (OR) (b) Verify Stoke’s theorem for a vector defined by F = y i+z j+xk , where S is the upper half of the surface of the sphere x2 + y2 + z2 = 1 and C is its boundary. ................
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