Differential Equations — Questions & Answers

Sample questions from Differential Equations

1. Q1. The number of arbitrary constants in the particular solution of a DE is:

   • A.Equal to order
   • B.Less than order
   • C.Zero✓
   • D.1
   Solution

   C is correct. A particular solution has all arbitrary constants determined by initial/boundary conditions, so it contains ZERO arbitrary constants (all are fixed).

2. Q2. The order of the DE obtained by eliminating a, b from y = ae^x + be^(2x) is:

   • A.1
   • B.2✓
   • C.3
   • D.0
   Solution

   B is correct. y=ae^x+be^(2x) has 2 arbitrary constants. Eliminating both requires differentiating twice, yielding a 2nd-order ODE.

3. Q3. The degree of the differential equation (dy/dx)³ + 2y = 0 is:

   • A.1
   • B.2
   • C.3✓
   • D.0
   Solution

   C is correct. Degree = power of the highest order derivative when the equation is a polynomial in derivatives. Here dy/dx is raised to power 3, so degree = 3.

4. Q4. The solution of the differential equation (dy/dx)^2 - 5(dy/dx) + 6 = 0 is:

   • A.y = -2x + C only
   • B.y = 5x + C
   • C.(y - 2x - C)(y - 3x - C) = 0✓
   • D.y = 6x + C
   Solution

   Factorising, (dy/dx - 2)(dy/dx - 3) = 0, so dy/dx = 2 or dy/dx = 3. Integrating each gives y = 2x + C or y = 3x + C, and the combined solution is (y - 2x - C)(y - 3x - C) = 0.

5. Q5. For the equation y dx + (x - y^3) dy = 0, the correct observation is that it is:

   • A.separable as written
   • B.homogeneous of degree 3
   • C.linear in x with integrating factor y, as dx/dy + x/y = y^2✓
   • D.solvable with integrating factor e^y
   Solution

   Dividing by y dy gives dx/dy + x/y = y^2, which is linear in x with P(y) = 1/y and integrating factor y. Then d/dy(xy) = y^3, so xy = y^4/4 + C.

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