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Comprehensive Problem Bank for MA 108 (Formerly Part of MA 203)

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Comprehensive Problem Bank for MA 108 (Formerly Part of MA 203) Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai{400 076, INDIA. MA 203{Mathematics III Autumn 2006 Instructors: A. Athavale N. Nataraj R. Raghunathan (Instructor in-charge) G. K. Srinivasan Name : Roll No : Course contents of MA 203 (Mathematics III): Ordinary differential equations of the 1st order, exactness and integrating factors, variation of pa- rameters, Picard's iteration method. Ordinary linear differential equations of the nth order, solution of homogeneous and non-homogeneous equations. The operator method. The methods ofundetermined co- efficients and variation of parameters. Systems of differential equations, Phase plane. Critical points, stability. Infinite sequences and series of real and complex numbers. Improper integrals. The Cauchy criterion, tests of convergence, absolute and conditional convergence. Series of functions. Improper integrals depend- ing on a parameter. Uniform convergence. Power series, radius of convergence. Power series methods for solutions of ordinary differential equations. Legendre equation and Legendre polynomials, Bessel equations and Bessel functions of first and second kind. Orthogonal sets of functions. Sturm-Liouville problems. Or- thogonality of Bessel functions and Legendre polynomials. The Laplace transform. The Inverse transform. Shifting properties, convolutions, partial fractions. Fourier series, half-range expansions. Approximation by trignometric polynomials. Fourier integrals. Texts/References E. Kreyszig, Advanced Engineering Mathematics, 8th ed., Wiley Eastern, 1999. Teaching Plan [K] refers to the text book by E. Kreyszig, \Advanced Engineering Mathematics", 8th Edition, John Wiley and Sons(1999). Policy for Attendance Attendance in both lectures and tutorial classes is compulsory. Students who fail to attend 80% of the lectures and tutorial classes may be awarded an XX grade. Evaluation: Figures in parentheses denote the percentage of the marks assigned to each quiz or exam. Quiz 1 (16.67): 6:00 - 7:00 p.m., Wednesday, August 16. Quiz 2 (16.67): 6:00 - 7:00 p.m., Wednesday, October 18 Mid-semester examination (33.33) End-semester examination (33.33) 1 Topics to be covered before the mid-semester examination No. Topic in [K] No.of Lec. x 1. Basic concepts and ideas, 1.1-1.2 1st order linear equations (homogeneous and non homogeneous), Separation of Variables, 1.3 - 1.4 3 Exact equations, integrating factors The Bernoulli Equation 1.5-1.7 2. Existence and Uniqueness: Picard's iteration 1.9 Singular solutions, enveloppes and 4 orthogonal trajectories 1.8 3. Linear Differential equations: Homogeneous equations with constant coefficients, 2.1-2.3 4 Existence and Uniqueness, The Wronskian, Non-homogeneous equations 2.7-2.8 4. The Cauchy-Euler equations The Method of undetermined coefficients, 2.9{2.10 4 The method of variation of parameters 2.11-2.12 2 5. Sequences and Series, Convergence Tests 14.1, A- 3.3 3 6. Uniform convergence, and Power Series 14.2 - 14.5 3 Topics to be covered after the mid-semester examination No. Topic in [K] No.of Lec. x 7. Improper integrals (p. 27 of this booklet) 3 8. Laplace transforms and systems of equations 5.1-5.7 4 9. Series solutions: Legendre's equation and the Legendre polynomials 4.3 3 10. Frobenius' method, the Bessel's functions 4.4{4.6 4 10. Sturm-Liouville Problems: Eigenvalues and eigenfunctions properties and more 4.7-4.8 3 11. Fourier Series and Fourier Integrals 10.1 - 10.10 4 12. Qual. Theory: Phase plane analysis, 3.3 - 3.5 Critical Points, Linearized Stability. 3 2 Tutorial Sheet No. 1 Q.1. Classify the following equations (order, linear or non-linear): d3y dy 2 dy d2y dy (i) dx3 + 4( dx ) = y (ii) dx + 2y = sin x (iii) y dx2 + 2x dx + y = 0 4 2 6 d y dy 2 2 d y d y t (iv) dx4 + (sin x) dx + x y = 0: (v)(1 + y ) dt2 + t dt6 + y = e : Q.2. Formulate the differential equations represented by the following functions by eliminating the arbi- trary constants a; b and c: (i) y = ax2 (ii) y a2 = a(x b)2 (iii) x2 + y2 = a2 (iv) (x a)2 + (y b)2 = a2 − − − − (v) y = a sin x + b cos x + a (vi) y = a(1 x2) + bx + cx3 (vii) y = cx + f(c): − Also state the order of the equations obtained. Q.3. Solve the equation x3(sin y)y = 2. Find the particular solution such that y(x) π as x + . 0 ! 2 ! 1 Q.4. Prove that a curve with the property that all its normals pass through a point is a circle. Q.5. Find the values of m for which (a) y = emx is a solution of (i) y + y 6y = 0 (ii) y 3y + 2y = 0: 00 0 − 000 − 00 0 (b) y = xm for x > 0 is a solution of (i) x2y 4xy + 4y = 0 (ii) x2y xy + y = 0: 00 − 0 000 − 00 0 Q.6. For each of the following linear differential equations verify that the function given in brackets is a solution of the differential equation. x x (i) y00 + 4y = 5e + 3 sin x (y = a sin 2x + b cos 2x + e + sin x) (ii) y 5y + 6y = 0, (y = e3x; y = e2x; c y + c y ) 00 − 0 1 2 1 1 2 2 (iii) y + 6y + 11y + 6y = e 2x (y = ae x + be 2x + ce 3x xe 2x) 000 00 0 − − − − − − (iv) y + 8y = 9ex + 65 cos x, (y = ae 2x + ex(b cos p3x + c sin p3x) + 8 cos x sin x + ex) 000 − − Q.7. Let 'i be a solution of y0 + ay = bi(x) for i = 1; 2. Show that '1 + '2 satisfies y0 + ay = b1(x) + b2(x). Use this result to find the solutions of y0 + y = sin x + 3 cos 2x passing through the origin. Q.8. Obtain the solution of the following differential equations: 2 2 (i) (x + 1)dy + (y + 4)dx = 0; y(1) = 0 (ii) y0 = y cot x; y(π=2) = 1 (iii) y = y(y2 1), with y(0) = 2 or y(0) = 1, or y(0) = 0 (iv) (x + 2)y xy = 0; y(0) = 1 0 − 0 − y x 2 (v) y0 + − = 0; y(1) = 1 (vi) y = (y x) ; y(0) = 2 y + x 0 − 1 (vii) 2(y sin 2x + cos 2x)dx = cos 2xdy; y(π) = 0. (viii) y0 = (x + 1)(x2 + 1) 3 Q.9. For each of the following differential equations, find the general solution (by substituting y = vx) 2 y xy 2 2 2 (i) y0 = − (ii) x y0 = y + xy + x x2 + xy 0 2 (iii) xy = y + x cos (y=x) (iv) xy0 = y(ln y ln x) − dy ax + by + m Q.10. Show that the differential equation = where a; b; c; d; m and n are constants can be dx cx + dy + n dy ax + by reduced to = if ad bc = 0: Then find the general solution of dt cx + dy − 6 (i) (1 + x 2y) + y0(4x 3y 6) = 0 y −x+1 − − (ii) y0 = y−x+5 − (iii) (x + 2y + 3) + (2x + 4y 1)y = 0. − 0 1 Q.11. Solve the differential equation 1 y2dx+p1 x2dy = 0 with the conditions y(0) = p3. Sketch − − 2 the graphs of the solutions andpshow that they are each arcs of the same ellipse. Also show that after these arcs are removed, the remaining part of the ellipse does not satisfy the differential equation. Q.12. The differential equation y = xy0 + f(y0) is called a Clairaut equation (or Clairaut's equation). Show that the general solution of this equation is the family of straight lines y = cx + f(c). In addition to these show that it has a special solution given by f (p) = x where p = y . This special solution 0 − 0 which does not (in general) represent one of the straight lines y = cx + f(c), is called a singular solution. Hint: Differentiate the differential equation. Q.13. Determine the general solutions as well as the singular solutions of the following Clairaut equations. In each of the two examples, sketch the graphs of these solutions. 02 (i) y = xy0 + 1=y0. (ii) y = xy0 y0= 1 + y − p Q.14. For the parabola y = x2 find the equation of its tangent at (c; c2) and find the ordinary differential equation for this one parameter family of tangents. Identify this as a Clairaut equation. More generally take your favourite curve and determine the ODE for the one parameter family of its tangents and verify that it is a Clairut's equation. N.B: Exercise 13 shows that the converse is true. Q.15. In the preceeding exercises, show that in each case, the envelope of the family of straight lines is also a solution of the Clairaut equation. 3 3=2 Q.16. Show that the differential equation y0 y = 2x− has three distinct solutions of the form A=px − but that only one of these is real valued. Tutorial Sheet No. 2 Q.1. State the conditions under which the following equations are exact. (i) [f(x) + g(y)]dx + [h(x) + k(y)]dy = 0 (ii) (x3 + xy2)dx + (ax2y + bxy2)dy = 0 (iii) (ax2 + 2bxy + cy2)dx + (bx2 + 2cxy + gy2)dy = 0 4 Q.2. Solve the following exact equations (i) 3x(xy 2)dx + (x3 + 2y)dy = 0 (ii) (cos x cos y cot x)dx sin x sin ydy = 0. − − − (iii) exy(x + y)dx + ex(x + 2y 1)dy = 0 − Q.3. Determine (by inspection suitable) Integrating Factors (IF's) so that the following equations are exact. (i) ydx + xdy = 0 (ii) d(ex sin y) = 0 y (iii) dx + ( )2dy = 0 (iv) yex=ydx + (y xex=y)dy = 0 x − (v) (2x + ey)dx + xeydy = 0, (vi) (x2 + y2)dx + xydy = 0 Q.4.
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