Grade 12 : Mathematics | Differential Equations

Question 1

What are the order and degree (if defined) of the differential equation $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0$?

Accepted Answer

C) Order = 2, Degree is not defined

Answer

A) Order = 2, Degree = 3

Answer

B) Order = 1, Degree = 2

Answer

D) Order = 3, Degree = 2

Question 2

The integrating factor of the differential equation $x \frac{dy}{dx} - y = 2x^2$(for $x > 0$) is:

Accepted Answer

B) $\frac{1}{x}$

Answer

A) $x$

Answer

C) $e^{-x}$

Answer

D) $-x$

Question 3

What is the general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$?

Accepted Answer

A) $e^{-y} + e^x = C$

Answer

B) $e^y - e^x = C$

Answer

C) $e^{-y} - e^x = C$

Answer

D) $e^y + e^x = C$

Question 4

The number of arbitrary constants in the general solution and the particular solution of a differential equation of third order are respectively:

Accepted Answer

C) 3 and 0

Answer

A) 3 and 3

Answer

B) 0 and 3

Answer

D) 2 and 1

Question 5

Assertion (A): The function $y = A\cos x + B\sin x$ is a solution of $\frac{d^2y}{dx^2} + y = 0$.
Reason (R): The degree of the differential equation $\frac{d^2y}{dx^2} + y = 0$ is 1.

Select the correct choice from the following:

Accepted Answer

B) Both (A) and (R) are true but (R) is not the correct explanation of (A).

Answer

A) Both (A) and (R) are true and (R) is the correct explanation of (A).

Answer

C) (A) is true but (R) is false.

Answer

D) (A) is false but (R) is true.

Question 6

Which of the following functions $f(x, y)$ is a homogeneous function of degree 0?

Accepted Answer

A) $\frac{x - y}{x + y}$

Answer

B) $\frac{x^2 - y^2}{x + y}$

Answer

C) $x^2 + y^2$

Answer

D) $\frac{x^3 + y^3}{x - y}$

Question 7

Find the general solution of the differential equation $\frac{dy}{dx} + y = 1$ ($y eq 1$).

Accepted Answer

D) $y = 1 + Ce^{-x}$

Answer

A) $y = 1 + Ce^x$

Answer

B) $y = 1 - Ce^x$

Answer

C) $y = Ce^{-x} - 1$

Question 8

A chemical researcher models the rate of decomposition of a certain substance in an aqueous solution. The concentration $C(t)$ decreases at a rate directly proportional to the concentration itself, giving the differential equation $\frac{dC}{dt} = -kC$ where $k > 0$ is a constant.

If the initial concentration is $C_0$ at $t = 0$, what is the concentration $C(t)$ at any time $t$?

Accepted Answer

B) $C(t) = C_0 e^{-kt}$

Answer

A) $C(t) = C_0 - kt$

Answer

C) $C(t) = C_0 e^{kt}$

Answer

D) $C(t) = \frac{C_0}{1 + kt}$

Question 9

The particular solution of $\frac{dy}{dx} = \frac{2x}{y}$ satisfying the initial condition $y(0) = 2$ is:

Accepted Answer

A) $y^2 - 2x^2 = 4$

Answer

B) $y^2 + 2x^2 = 4$

Answer

C) $y^2 - x^2 = 4$

Answer

D) $y - x^2 = 2$

Question 10

The differential equation of the form $\frac{dx}{dy} + P_1 x = Q_1$ has the integrating factor given by:

Accepted Answer

C) $e^{\int P_1 dy}$

Answer

A) $e^{\int P_1 dx}$

Answer

B) $e^{\int Q_1 dy}$

Answer

D) $e^{-\int P_1 dy}$

Question 11

What substitution is standard to solve a homogeneous differential equation of the form $\frac{dy}{dx} = F\left(\frac{y}{x}\right)$?

Accepted Answer

B) $y = vx$

Answer

A) $x = vy$

Answer

C) $y = v + x$

Answer

D) $xy = v$

Question 12

Assertion (A): The differential equation $\left(\frac{dy}{dx}\right)^2 + y = \sin x$ has order 1 and degree 2.
Reason (R): The order of a differential equation is the order of the highest derivative occurring in it, and the degree is the highest power of the highest order derivative when expressed as a polynomial in derivatives.

Select the correct choice:

Accepted Answer

D) Both (A) and (R) are true and (R) is the correct explanation of (A).

Answer

A) Both (A) and (R) are true and (R) is NOT the correct explanation of (A).

Answer

B) (A) is false but (R) is true.

Answer

C) (A) is true but (R) is false.

Question 13

The integrating factor for the differential equation $\cos x \frac{dy}{dx} + y \sin x = 1$ is:

Accepted Answer

A) $\sec x$

Answer

B) $\cos x$

Answer

C) $	an x$

Answer

D) $\sin x$

Question 14

Find the general solution of the differential equation $\frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}$.

Accepted Answer

B) $	an^{-1} y - 	an^{-1} x = C$

Answer

A) $	an^{-1} x - 	an^{-1} y = C$

Answer

C) $	an^{-1} y + 	an^{-1} x = C$

Answer

D) $\sin^{-1} y - \sin^{-1} x = C$

Question 15

In an electrical RL series circuit, the current $i(t)$ obeys the first-order differential equation $L \frac{di}{dt} + R i = E_0$, where $L, R$ and $E_0$ are positive real constants.

What is the integrating factor of this differential equation when solved for $i(t)$?

Accepted Answer

C) $e^{\frac{R}{L}t}$

Answer

A) $e^{\frac{L}{R}t}$

Answer

B) $e^{-\frac{R}{L}t}$

Answer

D) $\frac{R}{L}t$

Question 16

The general solution of the differential equation $x \, dy - y \, dx = 0$ represents a family of:

Accepted Answer

A) Straight lines passing through the origin

Answer

B) Circles centered at the origin

Answer

C) Parabolas with vertex at the origin

Answer

D) Hyperbolas asymptotic to the axes

Question 17

The solution of the differential equation $\frac{dy}{dx} + y\cot x = 2\cos x$ has integrating factor:

Accepted Answer

D) $\sin x$

Answer

A) $\cos x$

Answer

B) $\cot x$

Answer

C) $\csc x$

Question 18

What is the sum of the order and the degree of the differential equation $1 + \left(\frac{dy}{dx}\right)^3 = \frac{d^2y}{dx^2}$?

Accepted Answer

B) 3

Answer

A) 4

Answer

C) 5

Answer

D) 2

Question 19

Solve the differential equation $\frac{dy}{dx} = \frac{y}{x} + 1$(for $x > 0$).

Accepted Answer

A) $y = x\ln x + Cx$

Answer

B) $y = \ln x + C$

Answer

C) $y = x^2 + Cx$

Answer

D) $y = x\ln x + C$

Question 20

The solution of $(1 + x^2)dy + 2xy\,dx = 0$ with initial condition $y(0) = 1$ is:

Accepted Answer

C) $y(1 + x^2) = 1$

Answer

A) $y(1 + x^2) = 2$

Answer

B) $y = 1 + x^2$

Answer

D) $y = (1 + x^2)^2$

Set 5Grade 12 : Mathematics | Differential Equations