BSc Mathematics SEM IV 2018 19 Apr 2018-19 MATHEMATICS PAPER II 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 A is considered to be ordinary if it has 20 marks
   • (a) more than one dependent (b) one independent variable
   • (c) more than one independent (d) None of these The order and degree of the differential equation 5xy is
   • (a) 2and1
   • (b) 2and2
   • (c) land2
   • (d) land1 The function f(x,y) = — 7xy + tan (2)
   • (a) is homogenous of degree | is homogenous of degree 2
   • (c) is homogenous of degree 3 (d) not homogenous
   • iv. The differential equation 2x — y = a family of
   • (a) lines (b) circles
   • v. General solution of =0 isy =
   • (c) sin 2x + cos 2x (d)
   • vi. General solution for differential equation — y'= 0 is vil. Wronskian determinant with usual symbols is equal to
   • viii. One of the solutions of the homogeneous linear system of differential equations
   • (c) x = (d) None of these
   • ix. The Wronskian of two solutions &(2(t), y2(t)) for the linear system of first order homogeneous differential equations is
   • (c) (d) None of these One of the solutions of the homogeneous linear system iS
   • (c) (d) None of these Attempt any ONE question from the following: (08)
   • a) Show that the general solution of the linear first order O.D.E. + Py = Q, where P and Q are integrable functions of x, = f Qe! + c), c being an arbitrary constant. Hence solve the O.D.E. + 2xy
   • ii. The current i(t) at time t in an electrical circuit containing a source of inductance and resistance is governed by the differential equation L < E(t), where the inductance L and the resistance R are constant whereas the E(t) is a function of time t. Determine the current
   • (a) If the initial current is 0 and the applied is constant
   • (b) If the initial current is 0 and the applied is periodic in time t and given as E(t) = cos wt , where and w are constants Attempt any TWO questions from the following: (12)
   • b) i. Show that the following differential equation is non-exact. Hence find the I.F. and Solve the following Bernoulli’s Differential equation. =y Find the orthogonal trajectories of AY =X
2. Q3 Attempt any ONE question from the following: 8 marks
   • a) i. Let m, and m, be the roots of the auxiliary equation of the differential equation = 0, where p and q are constants. Discuss the general solution of the differential equation when
   • (a)m, and are real and unequal
   • (b)m, and are complex roots
   • ii. Let y,(x)be a non-zero solution to the differential equation P(x)y + Q(x)y = 0 on [a,b]. Then show that another linearly independent solution
3. Q3 Attempt any TWO questions from the following: 12 marks
   • b) i. Find the general solution for the differential equation — 5y + 6y = 0
4. Q1 Solve the differential equation +3 = 6e the method of variation of parameters solve + 4y = cosx
   • iv. Show that y(x) = + is solution for the equation — + 2y = 0, hence find particular solution,if y(1) = 3, =5
5. Q4 a) Attempt any ONE question from the following: that the two solutions and of the homogeneous linear system 4 gy are linearly dependent on [a, b] iff their Wronskian is identically zero on [a, b] 8 marks
   • ii. What do we mean by the general solution of a system of linear homogeneous
   • O.D.E. of the first order in two variables? Let Jand two solutions of the following homogeneous linear system dy on [a, b]. Prove that is also a solution for any real constants c, and
6. Q4 b) Attempt any TWO questions from the following: Solve the linear system: Find the general solution of the following linear system: Wronskian of the two solutions and of the homogeneous system dy . Show that this Wronskian is either identically zero or nowhere zero on [a, b] 12 marks
   • iv. Show that (—2e‘ sin 2t, cos 2t) and cos 2t sin 2t) are linearly independent solutions of QS. Attempt any FOUR questions from the following: (20)
   • a) Check whether the following differential equations are exact and solve
   • c) Show that y = is a solution ofx*y" + 2y = 0 on any interval not containing the origin
   • d) Solve the differential equation + y = x by the method of variation of parameters Find the general solution of the system:
   • f) Show that bothx,; = y, = and x, = —e ‘are solutions of the system . Also show that these two solutions are linearly independent

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