RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

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1 RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral: (D + 2)(D ) 2 y = e 2x. 5. Find Particular Integral: (D 3 + )y = cos(2x ). 6. Find Particular Integral: (D 2 + D)y = x 2 + 2x Reduce to linear differential equation x 2 d2 y x dy dx 2 dx + y = log x. 8. Reduce to linear differential equation x d2 y dx 2 2y x = x + x Reduce to linear differential equation ( + x) 2 d2 y + ( + x)x dy dx 2 dx + y = 2 sin x. 0. Reduce to linear differential equation (2x+3) 2 d2 y (2x+3) dy dx 2 dx 2yy = 6x.. Solve d2 y dx 2 4y = x sinh x. PART B 2. Solve (D 2 )y = x sin 3x + cos x. 3. Solve (D 2 4D + 3)y = sin 3x cos 2x. 4. Solve d2 y dx 2 + a 2 y = sec ax. 5. Solve x 2 d2 y dx 2 + 4x dy dx + 2y = ex. 6. Solve x 3 d3 y dx 3 + 2x 2 d2 y dx 2 + 2y = 0(x + x ). 7. Solve ( + x) 2 d2 y + ( + x) dy dx 2 dx + y = sin[2 log( + x)]. 8. Solve (2 + 3x) 2 d2 y dx 2 + 3(2 + 3x) dy dx 36y = 3x2 + 4x Solve dx dt t = 0. dy + y = sin t; dt + x = cos t Given that x = 2 and y = 0 when 0. Solve Dx + Dy + 3x = sin t; Dx + y x = cos t QUESTION BANK MATHEMATICS II

2 RAJALAKSHMI ENGINEERING COLLEGE MA 26. Find.( r r ) UNIT II - VECTOR CALCULUS 2. Prove that (r n ) = nr n 2 r PART A 3. Find the unit normal vector to the surface x 2 + xy + z 2 = 4 at the point (,, 2). 4. What is the greatest rate of increase of φ = xy 2 z at (, 0, 3)? 5. The temperature at a point (x, y, z) in space is given by T (x, y, z) = x 2 + y 2 z. A mosquito located at (4, 4, 2) describes to fly in such a direction that it gets cooled faster. Find the direction in which it should fly. 6. If φ = yz i + xz j + xy k, then find φ. 7. Write down φ in orthogonal curvilinear co-ordinates. 8. For what value of k is the vector r k r solenoidal? 9. Determine f(r) so that the vector f(r) r is solenoidal. 0. Find a such that F = (3x 2y+z) i +(4x+ay z) j +(x y+2z) k PART B. If r is the position vector of the point P (x, y, z), prove (r n ) = nr n 2 r where r = r. 2. Find the directional derivative of φ = xy 2 z 3 at the point (,, ) along the normal to the surface x 2 + xy + z 2 = 3 at the point (,, ). 3. Find the directional derivative of φ = 3x 2 +2y+z 2 at the point (,, ) in the direction 2 i + 2 j k. 4. Find a and b such that the surface ax 3 by 2 z = (a + 3)x 2 and 4x 2 y z 3 = may cut orthogonally at (2,, 3). 5. Prove that divcurl F = Show that F = (6xy +z 3 ) i +(3x 2 z) j +(3xz 2 y) k is irrotational vector and find the scalar potential function F = φ. QUESTION BANK 2 MATHEMATICS II

3 RAJALAKSHMI ENGINEERING COLLEGE MA Apply Green s Theorem in the plane to evaluate C (3x2 8y 2 )dx + (4y 6xy)dy where C is the boundary of the region defined by x = 0, y = 0, x + y =. 8. Apply Green s Theorem in the plane to evaluate C (3x2 8y 2 )dx + (4y 6xy)dy where C is the boundary of the region defined by y = x, y = x Verify Gauss s divergence theorem for F = 4xz i y 2 j + yz k taken over the cube bounded by x = 0, x =, y = 0, y =, z = 0, z =. 0. Verify Stoke s theorem for F = (x 2 y 2 ) i 2xy j taken over the cube bounded by x = ±a, x =, y = 0, y = b. QUESTION BANK 3 MATHEMATICS II

4 RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT III - ANALYTIC FUNCTIONS PART A. What is the necessary conditions for the existence of the derivative of f(z)? 2. If w = log z, then determine where w is non analytic. 3. Define singular point of the function with an example. 4. Show that f(z) = xy + iy is not analytic. 5. Define holomorphic function. 6. Show that v = e x sin y is harmonic function. 7. Show that v = e x cos y is harmonic function. 8. Find the invariant points of the transformation w = z z+. 9. Under the transformation w = z, find the image of z 2i = Show that the transformation w = z transforms all circles and straight lines to circles and straight lines in the w-plane.. State two important properties of Mobius transformation. PART B. If f(z) is a regular function of z prove that 4 f (z) If f(z) is an analytic function of z, prove that 0. ( ) f(z) x 2 y 2 = 2 ( ) log f(z) = x 2 y 2 3. Show that the function f(z) = xy is not analytic at the origin even though C-R equations are satisfied. 4. Determine the analytic function whose real part is sin 2x (cosh 2y cos 2x). 5. Determine the analytic function whose real part is e 2x (x cos 2y y sin 2y). 6. Determine the analytic function whose imaginary part is e x sin y. QUESTION BANK 4 MATHEMATICS II

5 RAJALAKSHMI ENGINEERING COLLEGE MA Determine the analytic function whose imaginary part is e x (x sin y y cos y). 8. find the analytic function z = u + iv, if u v = x y x 2 +4xy+y 2 9. Find the bilinear transformation that maps the points + i, i, 2 i of the z-plane into the points 0,, i of the w-plane 0. Find the bilinear transformation that maps the points i,, of the z-plane into the points 0,, of the w-plane QUESTION BANK 5 MATHEMATICS II

6 RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT IV - COMPLEX INTEGRATION. State Cauchy s Theorem. PART A 2. Evaluate C (z a) dz where C is a simple closed curve and the point z = a is (i) outside C, (ii) inside C. 3. Prove that C (z a)n dz = 0, [n ] where C is the circle z a = r. 4. Find the Taylor s series expansion of z 2 in z <. 5. Find the Taylor s series expansion of sin z in about z = π Find the poles of f(z) = (z )2 z(z 2) Find the zeros of f(z) = (z2 +) 2 ( z 2 ). 8. What are the critical points of the transformation w = z + z. 9. State Cauchy s integral formula. 0. State Cauchy s Residue formula. PART B. Find the Laurent s series of f(z) z( z) valid in the region (i) z+ <, (ii) < z + < 2 and (iii) z + > 2. z 2. Find the Laurent s series of f(z) 2 (ii)2 < z < 3 and (iii) z > 3. z 2 +5z+6 valid in the region (i) z < 2, 3. Find the Taylor s series expansion of f(z) = State the region of convergence of the series. z 4. Find the Taylors series expansion of f(z) = state the region of convergence. z( z) (z+)(z+2) about z =. about z = i and 5. Using Cauchy s residue theorem evaluate C circle z 2 = Using Cauchy s residue theorem evaluate C circle z 2 = 2. zdz (z )(z 2) 2 where C is the (3z 2 +z)dz where C is the (z )(z 2 +9) QUESTION BANK 6 MATHEMATICS II

7 RAJALAKSHMI ENGINEERING COLLEGE MA Evaluate 8. Evaluate 0 9. Evaluate 0. Evaluate 0 x 2 dx using contour integration, where a > b > 0. (x 2 +a 2 )(x 2 +b 2 ) dx (x 2 +a 2 ) 3 using contour integration, where a > 0. cos xdx using contour integration, where a > b > 0. (x 2 +a 2 )(x 2 +b 2 ) x sin xdx (x 2 +a 2 ) using contour integration. QUESTION BANK 7 MATHEMATICS II

8 RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT V - LAPLACE TRANSFORM. Define Laplace transform. PART A 2. State the change of scale property in Laplace transformation. 3. State the first shifting property in Laplace transformation. 4. State the second shifting property in Laplace transformation. 5. Find the Laplace of unit step function. 6. Find the Laplace of unit impulse function. 7. Find the Laplace transforms of (a) sin 2t sin 3t (b) cos 2 2t (c) sin 3 2t (d) e 3t 2 cos 5t 3 sin 5t (e) e 3t sin 2 t (f) t cos at (g) t 2 sin at (h) et t (i) t 3 e 3t (j) te t sin 3t (k) sin kt kt cos kt sin at (l) t (m) sin2 t t 8. Find L{f(t)}, if f(t) = 9. Find L( t) and L πt. 0. Find L (F (s)) if F (s) is : (a) (s+) 3 2 { sin 2t, 0 < t < 2π ; 0, t > π. QUESTION BANK 8 MATHEMATICS II

9 RAJALAKSHMI ENGINEERING COLLEGE MA 26 (b) s2 +2s+3 s 3 s (c) (s 4) 5 (d) (3s 4) 4 (e) 2s+3 s 2 +4 (f) s(s+a) (g) s 3 s 2 6s+0 (h) s 2 +2s+5 (i) log s+ s (j) cot s (k) log s2 + s Find the Laplace transforms of (a) e 3t (2t + 3) 3 (b) e t sin 3t cos t (c) te t cos t (d) te 2t sin 3t (e) t cos 2t (f) (t sin t) 2 (g) ( + te t ) 3 (h) cosh at cos at PART B 2. Find the inverse Laplace transforms of (a) s s 2 +4 (b) s 3 + (c) s 4 +4a 4 (d) s 2 s (e) s 2 + s 4 +s 2 + (f) s+ s 2 +s+ (g) log( + a2 s 2 ) QUESTION BANK 9 MATHEMATICS II

10 RAJALAKSHMI ENGINEERING COLLEGE MA 26 (h) tan (s 2 ) (i) tan ( s+2 3 ) 3. Use convolution theorem to find the inverse Laplace transforms of the following functions: (a) (b) (s+)(s+2) s (s 2 +a 2 ) 2 (c) s 2 (s 2 +a 2 ) 2 (s 2 +b 2 ) 2 (d) (s 4 +4) 4. Solve y 4y + 8y = e 2t, y(0) = 2, y (0) = 2 5. Solve y + 4y = sin ωt, y(0) = 0, y (0) = 0 6. Solve y + 9y = cos 2t, y(0) =, y( π 2 ) = 7. Solve the simultaneous differential equations dy dt + 2x = cos 2t, x(0) =, y(0) = 0. dy dt 8. Solve y + y = t cos 2t, y(0) = y (0) = 0 + 2x = sin 2t and 9. Solve y + 4y = cos 2t, y(π) = y (π) = 0 0. Solve x 2x + x = t 2 e t, x(0) = 2, x (0) = 3 QUESTION BANK 0 MATHEMATICS II