Q.1 Which of the following is a first-order differential equation?
y'' + y = 0
dy/dx + y = 0
d²y/dx² + 2y = 0
d³y/dx³ - y = 0
Explanation - A first-order differential equation involves only the first derivative of y. Here, dy/dx + y = 0 has only the first derivative.
Correct answer is: dy/dx + y = 0
Q.2 The equation dy/dx = 3x² is classified as:
Linear differential equation
Nonlinear differential equation
Second-order differential equation
Homogeneous differential equation
Explanation - Since dy/dx = 3x² can be written as dy/dx - 3x² = 0, it is linear in y.
Correct answer is: Linear differential equation
Q.3 The solution of dy/dx = y is:
y = Ce^x
y = Cx
y = x² + C
y = ln(x) + C
Explanation - Separating variables gives dy/y = dx. Integrating gives ln(y) = x + C → y = Ce^x.
Correct answer is: y = Ce^x
Q.4 What type of differential equation is dy/dx = x + y?
Linear first-order
Nonlinear first-order
Second-order
Separable
Explanation - The equation can be written as dy/dx - y = x, which is linear in y.
Correct answer is: Linear first-order
Q.5 Which of the following is a homogeneous differential equation?
dy/dx = (x+y)/x
dy/dx = x² + y
dy/dx + y = 5
dy/dx = sin(x)
Explanation - An equation is homogeneous if f(x,y) is a homogeneous function. (x+y)/x can be rewritten in terms of y/x.
Correct answer is: dy/dx = (x+y)/x
Q.6 The general solution of dy/dx = 0 is:
y = C
y = x + C
y = e^x
y = ln(x) + C
Explanation - Since dy/dx = 0 means slope is zero, y is a constant function.
Correct answer is: y = C
Q.7 If dy/dx = 2x, then y = ?
x² + C
2x² + C
ln(x) + C
e^x + C
Explanation - Integrating dy/dx = 2x gives y = x² + C.
Correct answer is: x² + C
Q.8 Which method is commonly used to solve dy/dx = ky?
Separation of variables
Integration by parts
Substitution
Partial fractions
Explanation - dy/dx = ky can be solved by writing dy/y = k dx and integrating both sides.
Correct answer is: Separation of variables
Q.9 A second-order differential equation must contain:
dy/dx only
d²y/dx²
No derivatives
Integrals
Explanation - A second-order differential equation always involves the second derivative.
Correct answer is: d²y/dx²
Q.10 The equation y'' + y = 0 is:
First-order linear
Second-order linear homogeneous
Nonlinear
Partial differential equation
Explanation - y'' + y = 0 has second derivative, is linear, and has no external forcing term, hence homogeneous.
Correct answer is: Second-order linear homogeneous
Q.11 Which of the following is a particular solution of dy/dx = y, with y(0)=2?
y = 2e^x
y = e^x
y = Ce^x
y = 2x
Explanation - General solution is y = Ce^x. Using y(0)=2 gives C=2.
Correct answer is: y = 2e^x
Q.12 The differential equation for simple harmonic motion is:
y'' + y = 0
y'' - y = 0
y' + y = 0
y'' + y' = 0
Explanation - The standard form of SHM equation is y'' + ω²y = 0, for ω=1 it is y'' + y = 0.
Correct answer is: y'' + y = 0
Q.13 Which of these is a linear differential equation?
dy/dx + y = 0
dy/dx = y²
y'' = y² + x
y dy/dx = x
Explanation - Linear equations have dependent variable and derivatives of power 1 only. dy/dx + y = 0 is linear.
Correct answer is: dy/dx + y = 0
Q.14 In the equation dy/dx = f(x)g(y), the method of solving is:
Integrating factor
Laplace transform
Separation of variables
Series expansion
Explanation - Such equations are separable and solved by integrating ∫dy/g(y) = ∫f(x)dx.
Correct answer is: Separation of variables
Q.15 The general solution of dy/dx = cos(x) is:
y = sin(x) + C
y = cos(x) + C
y = e^x + C
y = ln(x) + C
Explanation - Integrating dy/dx = cos(x) gives y = sin(x) + C.
Correct answer is: y = sin(x) + C
Q.16 Which technique is suitable for solving linear first-order DE of form dy/dx + Py = Q?
Integrating factor method
Laplace transform
Fourier series
Separation of variables
Explanation - Linear first-order equations are solved by multiplying with integrating factor e^(∫Pdx).
Correct answer is: Integrating factor method
Q.17 Which of the following represents a partial differential equation?
∂u/∂x + ∂u/∂y = 0
dy/dx + y = 0
y'' + y = 0
dy/dx = sin(x)
Explanation - Partial differential equations involve partial derivatives with respect to more than one independent variable.
Correct answer is: ∂u/∂x + ∂u/∂y = 0
Q.18 The equation dy/dx = y tan(x) is solved by:
y = Csec(x)
y = Ccos(x)
y = Ce^(sin(x))
y = Ctan(x)
Explanation - dy/y = tan(x) dx → ln(y) = ∫tan(x) dx = -ln|cos(x)| → y = Csec(x).
Correct answer is: y = Csec(x)
Q.19 If y'' = 0, then general solution is:
y = C1 + C2x
y = e^x
y = sin(x)
y = x² + C
Explanation - Integrating y'' = 0 → y' = C1 → y = C1x + C2.
Correct answer is: y = C1 + C2x
Q.20 Laplace transform is a useful tool for solving:
Algebraic equations
Differential equations
Trigonometric identities
Logarithmic equations
Explanation - Laplace transforms convert differential equations into algebraic equations for easier solving.
Correct answer is: Differential equations
Q.21 The integrating factor of dy/dx + 2y = 0 is:
e^(2x)
e^(-2x)
2e^x
x²
Explanation - Integrating factor is e^(∫2dx) = e^(2x).
Correct answer is: e^(2x)
Q.22 The order of the differential equation y''' + y'' + y' + y = 0 is:
1
2
3
4
Explanation - The order of a differential equation is the highest derivative present, here y'''.
Correct answer is: 3
Q.23 The degree of differential equation (d²y/dx²)³ + dy/dx = 0 is:
1
2
3
None
Explanation - Degree is power of highest order derivative. Here, (d²y/dx²)³ has degree 3.
Correct answer is: 3
Q.24 Which of the following is an exact differential equation?
(2xy + y²)dx + (x² + 2xy)dy = 0
(y - x)dx + (x + y)dy = 0
y dx + x dy = 0
x dx + y dy = 0
Explanation - An equation M dx + N dy = 0 is exact if ∂M/∂y = ∂N/∂x. Here both are equal to 2x+2y.
Correct answer is: (2xy + y²)dx + (x² + 2xy)dy = 0
Q.25 The solution of dy/dx = x/y is:
y² = x² + C
y = Ce^x
y = x² + C
y = ln(x) + C
Explanation - dy/dx = x/y → y dy = x dx → ∫y dy = ∫x dx → y²/2 = x²/2 + C → y² = x² + C.
Correct answer is: y² = x² + C