AP ECET MATHEMATICS Multiple Choice Questions (MCQs)

136. Bernoulli’s equation is of the form:

a) \( \frac{dy}{dx} + P(x)y = Q(x) \)

b) \( \frac{dy}{dx} + P(x)y^n = Q(x) \)

c) \( \frac{dy}{dx} = P(x)y + Q(x) \)

d) \( \frac{dy}{dx} = P(x)y^2 + Q(x) \)

137. The solution of the linear differential equation \( \frac{dy}{dx} + P(x)y = Q(x) \) is:

a) \( y = e^{\int P(x)dx} \)

b) \( y = e^{\int Q(x)dx} \)

c) \( y = e^{\int P(x)dx} \cdot Q(x) \)

d) \( y = e^{\int Q(x)dx} + C \)

138. Which of the following is the general form of a nth-order linear differential equation with constant coefficients?

a) \( a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \dots + a_1 \frac{dy}{dx} + a_0 y = 0 \)

b) \( a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \dots + a_1 \frac{dy}{dx} = 0 \)

c) \( a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{dy}{dx} + a_0 y = Q(x) \)

d) \( \frac{d^n y}{dx^n} + \dots + a_1 \frac{dy}{dx} + a_0 y = 0 \)

139. What is the general solution of the non-homogeneous linear differential equation \( \frac{d^2 y}{dx^2} + 3 \frac{dy}{dx} + 2y = e^x \)?

a) \( y = C_1 e^{-x} + C_2 e^{-2x} + A e^x \)

b) \( y = C_1 e^x + C_2 e^{2x} + A e^x \)

c) \( y = C_1 e^{-x} + C_2 e^{x} + A e^x \)

d) \( y = C_1 e^x + C_2 e^{-x} + A e^x \)

140. The particular integral for the equation \( \frac{d^2 y}{dx^2} + y = \sin(ax) \) is:

a) \( A \cos(ax) + B \sin(ax) \)

b) \( A \cos(ax) - B \sin(ax) \)

c) \( A \cos(ax) + B \sin(ax) + C \)

d) \( A \cos(ax) + B \sin(ax) + D \)

141. The solution to the differential equation \( \frac{dy}{dx} = y \cos(x) \) is:

a) \( y = A e^{\sin(x)} \)

b) \( y = A e^{\cos(x)} \)

c) \( y = A \cos(x) \)

d) \( y = A e^{x \cos(x)} \)

142. Which of the following is the particular solution for \( \frac{d^2 y}{dx^2} + 4y = \cos(2x) \)?

a) \( A \cos(2x) + B \sin(2x) \)

b) \( A \cos(2x) + B \sin(2x) + C \)

c) \( \frac{1}{2} \cos(2x) \)

d) \( \frac{1}{5} \cos(2x) \)

143. The complementary function of the differential equation \( \frac{d^2 y}{dx^2} - 5 \frac{dy}{dx} + 6y = 0 \) is:

a) \( e^{3x} + e^{2x} \)

b) \( C_1 e^{3x} + C_2 e^{2x} \)

c) \( C_1 e^{-3x} + C_2 e^{-2x} \)

d) \( e^{5x} \)

144. The form of the particular integral for the function \( \cos(ax) \) in the equation \( \frac{d^2 y}{dx^2} + y = \cos(ax) \) is:

a) \( A \cos(ax) + B \sin(ax) \)

b) \( A \cos(ax) \)

c) \( A e^{ax} + B e^{-ax} \)

d) \( A \cos(ax) + B e^{ax} \)

145. For the equation \( \frac{d^2 y}{dx^2} + 3 \frac{dy}{dx} + 2y = \sin(ax) \), the particular solution is:

a) \( A \cos(x) + B \sin(x) \)

b) \( A e^x + B e^{-x} \)

c) \( A \cos(x) + B \sin(x) + C \)

d) \( A \cos(x) + B \sin(x) + D \)

146. The general solution for the equation \( \frac{d^2 y}{dx^2} + 3y = \sin(ax) \) is:

a) \( A \cos(ax) + B \sin(ax) + C \cos(x) \)

b) \( A \cos(ax) + B \sin(ax) + C \)

c) \( A e^x + B e^{-x} + C \cos(x) \)

d) \( A \cos(ax) + B \sin(ax) + D \)

147. For a differential equation with constant coefficients, the general solution is:

a) A combination of polynomial functions

b) A linear combination of exponential functions

c) A combination of trigonometric functions

d) A combination of logarithmic functions

148. The method used for solving a non-homogeneous linear differential equation is:

a) Substitution method

b) Variation of parameters

c) Integration factor

d) Laplace transform

149. The characteristic equation for the second-order homogeneous differential equation \( \frac{d^2 y}{dx^2} + p \frac{dy}{dx} + qy = 0 \) is:

a) \( r^2 + pr + q = 0 \)

b) \( r^2 + q = 0 \)

c) \( r^2 + p = 0 \)

d) \( r^2 - p = 0 \)

150. Which of the following is the solution for the differential equation \( \frac{d^2 y}{dx^2} + 4y = 0 \)?

a) \( C_1 \cos(2x) + C_2 \sin(2x) \)

b) \( C_1 e^{2x} + C_2 e^{-2x} \)

c) \( C_1 e^x + C_2 e^{-x} \)

d) \( C_1 \cos(x) + C_2 \sin(x) \)