BSc Mathematics SEM III 2022 2023 Oct 2023 MATHEMATICS III Question Paper - Mumbai University | munotes

Questions asked in this paper

• (ii) Figures to the right indicate marks

1. Q1 Attempt any ONE question from the following : 8 marks
   • a) i. The substitution x = reduces the Cauchy equation = 0 into the differential equation to the given Cauchy equation if and only if m is the root of the equation
   • ii. Let and y2(x) be two solutions to the differential equation to y" + P(x)y'+ Q(x)y = 0 Where are continuous on an interval (a,b) then y, and are linearly dependent on (a. b)
   • b) Attempt any TWO questions from the following: 12
   • i. Solve the differential equation: ((2x — + (2x —1)D =0
   • ii. Find the particular solution to the differential equation (D* + = 0;
   • iii. Verify = e?* and = are solution to differential equation y" — 3y'+ 2y = 0. Also verify y = + is the solution of any
2. Q2 Attempt any ONE question from the following : 8 marks
   • a) i. Define Wronskian. If x = x,(t), y = y,(t)and x = y = y2(t) are two linearly independent solutions to the homogenecus linear system = + b, (t)y, = + then in ja, b] the general solution to above system is X = + = + where are arbitrary
   • ii. Obtain the general solution of a Homogeneous Linear system (of two equations) with constant coefficients, when its Auxiliary equation has two real and equal roots
   • b) Attempt any TWO question from the following: 12
   • i. Using method of variation of parameter find a particular solution = = —3x + 2y + 2sint and hence find the general solution
   • ii. Find the general to the linear system < = 4x- = 8x — 6y
   • iii. Show that the solutions x = y = e* and x = = are linearly independent of the system of equation < =x+
3. Q3 Attempt any ONE question from the following : 8 marks
   • a) i. State the procedure of Picard’s method and hence using it find 4th approximation polynomial of the differential equationy’ = 2x + y — SYBSC- SEM III MATHEMATICS - 75 MARKS 2 HRS
   • ii. Solve the system of ordinary differential equations < =x+y- = 2x
   • b) Attempt any TWO question from the following: correct upto 4 decimal places for finding y(1.25) 12
   • ii. Using Taylor’s method find the polynomial of degree 4 that satisfies the differential equation y’ = — x*, y(1) = 1.5 and using this find the approximation value of y(1.1) ili, Use Runge-Kutta method of 4th order to estimate y(1.25) when y’ =
4. Q4 Attempt any THREE question from the following : 15 marks
   • a) Solve the following differential equation y” + 9y = 12sin2x
   • b) Find the general solution to the following differential equation x2y" + 3xy' +
   • c) Reduce the differential equation — 7y’ + 6y = 0
   • d) Define a system of homogeneous linear differential equations of order]. State the condition for two solutions (x,, y, ) and (X2, ) to be linearly independent. Also write the general solution
   • e) Estimate y(0.25) using Runge-Kutta method of 2" order with W2 = 2/3 for the differential equation y’ = = landh

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