Differential Equations BS 3 Semester/Term UOS — University of Sargodha 2019

University of Sargodha

BS 3rd Term Examination 2019

Subject: Computer Science Paper: Differential Equations (MATH:2215)

Time Allowed: 2:30 Hours Maximum Marks: 80

Note: Objective part is compulsory. Attempt any four questions from subjective part.

Objective Part (Compulsory)

Q.1. Write short answers of the following in 2-3 lines each on your answer sheet. (2*16)

i. Define partial differential equation.

ii. Define explicit solution.

iii. Determine whether the DE u dv + (v + uv - ve^u)du is linear in u.

iv. Verify that y = e^-x/2 is an explicit solution of the DE: 2y' + y =

0.

v. State the Newton's law of cooling and warming and give its mathematical model.

vi. Find the integrating factor of the linear equation 3 dy/dx + 12y = 4

vii. Write general form of homogeneous linear nth-order differential equation.

viii. Determine that the set of functions f1(x) = x, f2(x) = x - 1, f3(x) = x + 3 is linearly independent on the interval(-∞, ∞).

ix. Write the auxiliary equation, the roots and the corresponding general solution of DE:

y'' - 10y' + 25y = 0

x. What will be the assumed particular solution yp for g(x) = 3x^2 - 5 sin 2x + 7xe^x

xi. Verify that the differential operator 2D-1 annihilates the function y = 4e^x/2.

xii. Find the linear differential operator that annihilates the function y = x + 3xe^6x.

xiii. Write Maclaurin series for xe^3x in summation notation.

xiv. Write the standard form of first order linear differential equation.

xv. Write the general solution of the Bessel's equation: x^2y'' + xy' + (x^2 - 1/9)y =

0.

xvi. Define separable equation.

Subjective Part (4*12)

Q.2.

a) Solve the given IVP by using proper substitution.

dy/dx = (3x+2y)/(3x+2y+2), y(-1) = -1

b) Determine whether the given DE is exact. If exact, then solve it. (x^3 + y^3)dx + 3xy^2dy = 0

Q.3. A tank contains 200 liters of fluid in which 30 grams of salt are dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4 L/min; the well mixed solution is pumped out the same rate. Find the number A(t) of grams of salt in the tank at time t.

Q.4. Solve the given differential equation by the method of undetermined coefficients-superposition approach. y'' + y = 2x sinx

Q.5. Solve the given system of differential equations by systematic elimination.

dx/dt = 2x - y

dy/dt = x

Q.6. Find the power series solution of the DE y'' + x^2y' + xy = 0 about the ordinary point x =

0.

Q.7. Use the improved Euler's method to obtain a four decimal approximation of the value y(1.5) for the solution of the initial value problem y' = 2xy, y(1) = 1 by using h = 0.1.