Differential Equation BSCS 3rd TERM Past Paper 2016 UOS

Paper: Differential Equation (Math. 2215)
Time Allowed: 2½ Hours
Maximum Marks: 80

Objective Section (16 Questions × 2 Marks)

Q.1 Attempt the following:

1. Verify that:
   y = (6/5)·e^(–2x) is a solution of the equation:
   dy/dx + 20y = 24
2. Define:
   What is a mathematical model?
3. General Solution:
   Solve the equation:
   d⁴y/dx⁴ − 7·d²y/dx² − 18y = 0
4. Solve:
   dp/dt = p − p²
5. Define:
   What is an explicit solution of a differential equation?
6. Find k:
   Determine the value of k such that the equation:
   (y³ + ky⁴ − 20x²y³)·dy = 0
   is exact.
7. Define:
   What is a linear differential equation?
8. Solve:
   dy/dx = 5y
9. Define:
   What is an annihilator operator?
10. Check Linear Independence:
    Are the following functions linearly independent?
    f₁(x) = x, f₂(x) = x − 1, f₃(x) = x + 3
11. Define with Example:
    What is a homogeneous differential equation? Give an example.
12. Solve:
    dy/dx = 1 + e^(y + x)
13. Convert to Matrix Form:
    dx/dt = 4x − 7y
    dy/dt = 5x
14. Solve:
    x²·dy/dx = 1
15. Find y₂:
    Given that y₁ = x·sin(ln x) is a solution of:
    x²·d²y/dx² − x·dy/dx + 2y = 0,
    find the second solution y₂.
16. Verify Operator:
    Verify that the operator:
    D² + 3D − 10
    annihilates the function:
    y = e^(2x) + 3·e^(–5x)

Subjective Section (4 Questions × 12 Marks)

Q2. Solve the differential equation:

y” − 4y’ + 4y = e^[(x + 1)·3x]

Q3. Find the solution of:

(x² + 1)·y” + x·y’ − y = 0

Q4. Solve the non-homogeneous equation:

y” + 2y’ + y = sin(x) + 3·cos(2x)

Q5. Solve the system of differential equations:

dx/dt = 4x + 7y
dy/dt = –2y

Q6. Solve with initial conditions:

2x·y” + 3y’ = 0
y(1) = 5, y'(1) = 3

Q7. Solve the nonlinear equation:

(x + 1)²·dy/dx + 2xy − (2xy² − 1)·dy/dx = 0
Given: y(1) = 1

Link: Khan Academy – Differential Equations

This resource provides a comprehensive overview of differential equations, including explanations and practice problems.