BSc Mathematics SEM IV 2018 19 May 2018-19 MATHEMATICS PAPER III Question Paper - Mumbai University | munotes

Questions asked in this paper

• (ii) Figures to the right indicate marks for respective parts

1. Q1 Choose correct alternative in each of the following The order f differential equation y = x? & + 20 marks
   • i. e order and degree of differential equation y = — 7x)
   • (c) 2and5 (d)
   • ii. Orthogonal trajectories of the family of curves xy = k represents a family of
   • (c) Hyperbolas (d) Circles The integrating factor of
   • (c) (d) None of these
   • iv. Which of the following is an exact differential equation?
   • v. Which of the following functions are linearly independent?
   • vi. general solution of + 9y = Ois + (- (d) None of these vil. If = cos mx and = sin mx , value of Wronskian W (yj, y2) is
   • (c) 0 (d) None of these
   • viii. The Wronskian of two solutions and of a homogeneous linear system of differential equations, is equal to
   • (c) (d) None of these of the solutions of the system =y-x 2 = 3x+yis
   • (c) x=e',y=3e' (d) None ofthese
   • x. and are linearly independent solutions of = =x+3y, = 3x + y then particular solution for which x(0) = 5,
2. Q2 Attempt any ONE question from the following: 8 marks
   • a) i. When a differential equation M(x, y)dx + = 0 is said to be exact? State and prove the necessary and sufficient condition for the above differential equation to be exact
   • ii. to solve the differential equation + P(x)y = Q(x)y"(n # 0,1) where P(x), Q(x) are functions of x and
3. Q2 Attempt any TWO questions from the following: 12 marks
   • b) differential equation M(x, y)dx + N(x, y)dy = 0 is said to be homogeneous? Solve the homogenous differential equation
   • ii. The population of a village increases at a rate proportional to the population at that time. In a period of 10 years, the population grew from 10 thousand to 15 thousand. What will be the population after 20
   • iii. Find the solution to the differential equation
4. Q3 Attempt any ONE question from the following: 8 marks
   • a) i. Show that a second order homogenous differential equation y” + P(x)y' + Q(x)y = 0 is completely determined by two linearly independent solutions y,(x) and yz(x). However, the converse is not true, that is and are not uniquely determined by the
   • ii. Lety’ +ay’+b=0 differential equation with constant coefficients and let m* + = 0. be the corresponding auxiliary equation with roots m,and Discuss the general solution of the differential equation when:
   • (p)m,and are real and equal m, are complex
5. Q3 Attempt any TWO questions from the following: 12 marks
   • b) i. Show that y = + bx + 3 is the general solution of — + 2y = 6. Further find a particular solution of this
   • ii. whether the following functions are linearly dependent (on any interval not containing 0) by finding their Wronskian
   • (P) = =VX = x*logx
   • iii. Using method of variation of parameters find a particular integral of the
   • iv. Solve the differential equation y” + 3y’ — 10y = 6e* , by the method of undetermined coefficients
6. Q4 Attempt any ONE question from the following: Find the general solution of system ily where and bz are constants when the roots of auxiliary equation are Define Wronskian W (t)of the two solutions and the homogeneous system dy where (t), b2(t) are continuous functions on [a, b] Show that their Wronskian is either identically zero or nowhere zero 8 marks
7. Q4 Attempt any TWO questions from the following: 12 marks
   • b) i. x = = andx = = are linearly independent solutions to the homogeneous system = 2x + y. Hence find the general solution Find the general solution to the following system = 7x + 6y,
8. Q1 the particular solution to the following system = 2x — By,
   • iv. xX = = x2(t) Let and of the homogeneous system dy where (t), b2(t) are continuous functions on + ) is also a solution of the above system of equation for t [a, b]
9. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) Solve the differential equation (4x + 3y + 1)dx + 2y+1)dy =0
   • b) Find the orthogonal trajectories of y = where kis a parameter
   • d) another linearly independent solution for the differential equation — 2xy' + 2y = 0, given that y, = x is a solution
   • e) Showthatx y = andx = are solutions of the system
   • f) Define of homogeneous linear differential equations of order 1 State the condition for two solutions (x,, y,) and yz) to be linearly independent. Also write the general solution

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