BSc Mathematics SEM IV 2019 20 Oct 2019-20 ATKT MATHS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 All questions carry equal marks
2. Q3 Figures to right Indicate full marks of subquestion
3. Q1 Choose correct alternative in each of the following the Bernoulli differential equation reduce to variable separable
   • i)0 None of these
   • b) The order and degree of differential equation = + 22 ~ By
   • i)3 and2 ii) 2 and3 iii) 3 and iv) | and 3 of the following is homogenous differential equation
   • d) Orthogonal trajectories of the family of curves xy=k represents a family of
   • e) and linearly independent solutions of = = then Particular solution for which x(0)=5,y(0)=1 is
   • c) botha & b d) None of these
4. Q9 If x71 x1 and x then Wey
   • a) 0 None of these
   • h) The general solution of
   • b) d) None of these
   • i) A order first degree differential equation of the form tpy=Qy" known as Bernoulli different equation
   • a) First b) Second c) Third e) None of these uNe(x,y)dy=0 is an exact differential equation then wis constant b) Bernoulli constant c)Integrating factor d) None of these
5. Q2 A) Attempt any ONE of the following 8 marks
   • a) Prove that the Bernoulli equation is reduces to linear differential equation by the transformation z=y'"
   • b) The general solution of the linear differential equation +py=Q is pdx = f pdx dx +c prove the above result
   • (B) Attempt any two of the following
   • a) Find the time required for the sum of money to double itself at 5% p.a Sf
   • b) Use the substitution y=vx to solve =
   • d) Solve =
6. Q3 (A) Attempt Any ONE of the following. 8 marks
   • a) Let y, (x) and y2(x)bé any two solutions to the differential equation y” + py’ + Qy = 0 on the interval [a, b], then their wronskian w(y,,y2) = is identically zero if and only if y, (x), and y2(x) are linearly dependant on [a, b
   • b) Consider the differential equation y" + py’ + Qy = R — (1)p,Q constant and R ig function of x with cf = c,y + c2y2 where y, and y2 are solution of y" + py’ + Qy = 0 — (2) let is y = uy, +vy2 where u and v are functions of x then prove dx, v=f dx and W is wronskian
7. Q3 (B) Attempt any Two of the following . 12 marks
   • a) Solve y" — 5y’ — 6y = e3x given y(0) = = 1
   • b) Solve the IVP y’ + 12y' + 36y = 0,y(1) = = 0
   • c) Find the other linearly independent solution to the differential equation (1 — x2)y" — 2xy' + 2y = 0 given y,= x is a solution
   • d) If y" + py’ + Qy = 0 then prove that c, y, (x) + c2y2(x) is also a solution for any constant cl and c2
8. Q4 (A) Attempt any ONE of the following. 8 marks
   • a) If w(t) wronskian of two solutions x = x,(t)y = y,(t) and x = x2(t),y = v2(t) to the homogeneous linear system +b, = a,(t)x + b2(t)y then prove either w(t) is identically equal to zero or w(t) is never zero on fa, b]
   • b) If the auxiliary equation to the homogeneous system of equation with constant has real equal roots m,m then there exist non trivial constant such that x = = and x = (A2 + = (B2 + are linearly independent solutions to the system (1)
9. Q4 (B) Attempt any Two of the following 12 marks
   • a) that x = 3t — 2,y = + 3 is particular solution to the non homogencous
   • b) Find the third order differential equation whose equivalent system of linear equation is
   • c)Find the general solutions to the system = = 4x - 8x
   • d) Prove that x = + 2,y = 2t — 1 isa particular solution to the non
10. Q5 Actempt any Four of the following 20 marks
    • a) Find the general solution to the given system < =x
    • b) Find the general solution to the system satisfying initial conditions =x+
    • d)Solve the IVP — 12y = 3e5* with y(0) = y'(0) =
    • e) Solve = 2y with initial condition y(1) = 0
    • f) If (x + y)" is an integrating factor of + 6y)dx + +9y + 3x) = 0 then find n and solve the equation

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