BSc Sem III 2023 2024 2024 MATHS III Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 For Q.1, Q.2 and Q. 3 attempt any one subquestion (each 8 marks) from part (a), and any two subquestions (each 6marks) from part (b)
2. Q3 For Q.4, attempt any three. (each 5 marks) any one. [each 8Mks]
3. Q1 If the auxiliary equation of the nth order homogeneous linear differential equation boy” + + bay = 0 .*has n distinct real roots ., then prove that the general solution of (*)is given by y = + + ++. where 2)Let yp be a particular solution to the non homogeneous linear differential equation + by + + = R(X) let be the general solution to the corresponding homogeneous linear differential equation by + + + = 0 then prove that y = + is the general solution to the given non homogeneous
   • (b)Attempt any two. [each 6Mks]
4. Q1 Let fo, fp be n functions defined on | =[a,b] where each of these functions is continuously differentiable atleast (n-1) times on the Wronskian W fa, #0 for some then prove that fo, ., f,are linearly independent
5. Q2 Find the general solution to the differential equation (9D* + 6D? + D)y = 0
6. Q3 If k solutions to nth order linear homogeneous differential + (x) + + + = 0 then Prove that c,y, + C2V2 also a solution of given differential equation wherec;, cz,
7. Q2 (a)Attempt any one. [each
8. Q1 If w(t) is the Wronskian of two solutions t [a, b],x = = y,(t)and x = = y2(t) to the homogeneous linear system + b, (t)y,2 = a,(t)x + of differential equations then prove that either w(t) is identically equal to zero or w(t) is never zero
9. Q2 Lett [a,b],x = x,(t),y = x = y = y2(t)be two solutions to the homogeneous system of equations =a,(t)x = + then prove that x(t) = c,x,(t) + y(t) = + also a solution to the above(*) homogeneous system of equations for where arbitrary constants
   • (b)Attempt any two. [each 6Mks] 1)Find the general solution to the following system = = = 8x — 6y VCD/ SYBSC- SEM III - MATHEMATICS 75MARKS 2)Prove that x = 3t + 2,y = 2t — 1 is particular solution to the non homogeneous linear system 2,2 = 4x — 2y — 8t — 8 ,hence find general solution to above system 3)Find general solution to the following system =x- = 4x + Sy
10. Q3 (a)Attempt any one. [each 8Mks] 7 Derive Picard’s method formula to solve differential equation to find nth approximation for the differential equations y’ = f(x, y)with = yo 2)Solve boundary value problem for y” — y = 0 with y(0) = 0, y(1) = 1 with h=0.5
    • (b)Attempt any two. [each 6 Mks]
11. Q1 Apply Modified Euler’s method to estimate the values using 2 approximation for y’ = y(0) = 1and estimate y(0.25) taking step size h=0.25
12. Q2 Use Runge Kutta order method fo approximate y at x =
13. Q3 For differential equation y” — 8y’ + 7y = 10 with y(0) = —1,y(1) = —3. estimate the value of y(0.5) taking the step size as h=0.5 using finite difference method
14. Q4 Attempt any three. [each 5 marks
15. Q1 Find the particular integral for (D* + 4)y = sin 2x 2)Find the general solution to the following differential equation (D? + 2D* — 15D)y = 0 3)Find the equivalent system of first order equation for the initial value problem x 4)Show that x = 3t — 2,y = —2t + 3 is a particular solution to the non homogeneous system
16. Q5 Evaluate + + V12 to 4 significant digits and find absolute and relative error
17. Q6 Use Taylor series method to find the value of y at x = = given y(0) =

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