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1. A differential equation is an equation that involves:
A. An unknown function and its derivatives
B. Only constants
C. Only algebraic variables without derivatives
D. Matrices
Answer: A. An unknown function and its derivatives
Explanation: By definition, a differential equation relates a function to one or more of its derivatives.
2. The order of a differential equation is defined as the:
A. Order of the highest derivative present in the equation
B. Power of the highest derivative
C. Number of terms in the equation
D. Number of independent variables
Answer: A. Order of the highest derivative present in the equation
Explanation: Order refers specifically to how many times the function has been differentiated in the highest derivative term.
3. The degree of a differential equation is defined only when the equation is:
A. Expressible as a polynomial in its derivatives
B. Any equation involving derivatives, regardless of form
C. Always assumed to be 1
D. Undefined for every equation
Answer: A. Expressible as a polynomial in its derivatives
Explanation: Degree is defined only for equations that can be written as a polynomial equation in derivatives, after removing radicals and fractions involving derivatives.
4. A differential equation of the form dy/dx = f(x)g(y) is called a:
A. Variable separable differential equation
B. Linear differential equation
C. Homogeneous differential equation
D. Exact differential equation
Answer: A. Variable separable differential equation
Explanation: Such equations can be solved by separating the variables x and y onto opposite sides before integrating.
5. The general solution of a differential equation contains:
A. A number of arbitrary constants equal to the order of the equation
B. No arbitrary constants at all
C. Always exactly one constant regardless of order
D. An unrelated infinite number of constants
Answer: A. A number of arbitrary constants equal to the order of the equation
Explanation: The number of independent arbitrary constants in the general solution matches the order of the differential equation.
6. A first order linear differential equation has the standard form:
A. dy/dx + Py = Q, where P and Q are functions of x
B. y'' + y = 0 only
C. dy/dx = a constant only
D. y = mx + c
Answer: A. dy/dx + Py = Q, where P and Q are functions of x
Explanation: This is the recognized standard form for a first order linear differential equation in y.
7. The integrating factor (I.F.) of the linear differential equation dy/dx + Py = Q is given by:
A. e^(∫P dx)
B. e^(∫Q dx)
C. ∫P dx
D. P · Q
Answer: A. e^(∫P dx)
Explanation: Multiplying through by this integrating factor makes the left-hand side an exact derivative, allowing direct integration.
8. A homogeneous differential equation can typically be solved by substituting:
A. y = vx
B. y = x + c
C. y = 1/x
D. y = eˣ
Answer: A. y = vx
Explanation: This substitution converts a homogeneous equation in x and y into a variable-separable equation in v and x.
9. The particular solution of a differential equation is obtained by:
A. Using given initial or boundary conditions to determine the arbitrary constants
B. Completely ignoring initial conditions
C. Always setting all constants to zero
D. Differentiating the general solution again
Answer: A. Using given initial or boundary conditions to determine the arbitrary constants
Explanation: Substituting known initial or boundary values into the general solution allows the arbitrary constants to be evaluated.
10. Differential equations are widely used to mathematically model:
A. Real-world problems involving rates of change, growth, and decay
B. Only static geometric shapes
C. Only fixed, unchanging quantities
D. Operations on matrices
Answer: A. Real-world problems involving rates of change, growth, and decay
Explanation: Because they relate a quantity to its rate of change, differential equations naturally describe dynamic real-world processes.
11. The differential equation dy/dx = ky (with k a constant) generally models:
A. Exponential growth or decay
B. Simple linear growth only
C. Purely oscillatory motion
D. No real physical phenomenon
Answer: A. Exponential growth or decay
Explanation: This equation's solution, y = Ce^(kt), is the classic form describing exponential growth (k > 0) or decay (k < 0).
12. Forming a differential equation from a given family of curves primarily involves:
A. Eliminating the arbitrary constants through repeated differentiation
B. Simply adding new constants
C. Directly integrating the family of curves
D. Using matrices to solve for constants
Answer: A. Eliminating the arbitrary constants through repeated differentiation
Explanation: By differentiating the family of curves as many times as there are arbitrary constants, those constants can be eliminated to form the differential equation.
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