CBSE Class 12 Mathematics Differentials Equation MCQs Set 08

Question. The order of the differential equation \( \frac{d^4y}{dx^4} - \sin\left(\frac{d^2y}{dx^2}\right) = 5 \) is
(a) 4
(b) 3
(c) 2
(d) Not defined
Answer: (a) 4

Question. The order of the following differential equation \( \frac{d^3y}{dx^3} + x\left(\frac{dy}{dx}\right)^5 = 4 \log\left(\frac{d^4y}{dx^4}\right) \) is
(a) Not defined
(b) 3
(c) 4
(d) 5
Answer: (c) 4

Question. The order and degree (if defined) of the differential equation, \( \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 = x\sin\left(\frac{dy}{dx}\right) \), respectively are
(a) 2, 2
(b) 1, 3
(c) 2, 3
(d) 2, degree not defined
Answer: (d) 2, degree not defined

Question. The order and degree of the differential equation \( \left(1+3\frac{dy}{dx}\right)^2 = 4\frac{d^3y}{dx^3} \) respectively are
(a) 1, \( \frac{2}{3} \)
(b) 3, 1
(c) 3, 3
(d) 1, 2
Answer: (b) 3, 1

Question. What is the product of the order and degree of the differential equation \( \frac{d^2y}{dx^2}\sin y + \left(\frac{dy}{dx}\right)^3 \cos y = \sqrt{y} \)?
(a) 3
(b) 2
(c) 6
(d) Not defined
Answer: (b) 2

Question. Degree of the differential equation \( \sin x + \cos\left(\frac{dy}{dx}\right) = y^2 \) is
(a) 2
(b) 1
(c) Not defined
(d) 0
Answer: (c) Not defined

Question. If \( p \) and \( q \) are respectively the order and degree of the differential equation \( \frac{d}{dx}\left[\left(\frac{dy}{dx}\right)^3\right] = 0 \), then \( (p-q) \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. If \( m \) and \( n \) are the order and degree of the differential equation \( \frac{\left(\frac{d^2y}{dx^2}\right)^5 + 4\frac{\left(\frac{d^2y}{dx^2}\right)^3}{\frac{d^3y}{dx^3}} + \frac{d^3y}{dx^3}}{\frac{d^3y}{dx^3}} = x^2 - 1 \), then
(a) \( m = 3, n = 3 \)
(b) \( m = 3, n = 2 \)
(c) \( m = 3, n = 5 \)
(d) \( m = 3, n = 1 \)
Answer: (b) \( m = 3, n = 2 \)

Question. The order of differential equation corresponding to family of curve \( y = Ae^{2x} + Be^{-2x} \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The number of arbitrary constants in the general solution of the differential equation \( \frac{dy}{dx} + y = 0 \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. The general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \) is
(a) \( e^x + e^{-y} = C \)
(b) \( e^{-x} + e^{-y} = C \)
(c) \( e^{x+y} = C \)
(d) \( 2e^{x-y} = C \)
Answer: (a) \( e^x + e^{-y} = C \)

Question. The solution of the differential equation \( \frac{dx}{x} + \frac{dy}{y} = 0 \) is
(a) \( \frac{1}{x} + \frac{1}{y} = C \)
(b) \( \log x - \log y = C \)
(c) \( xy = C \)
(d) \( x + y = C \)
Answer: (c) \( xy = C \)

Question. The number of solutions of the differential equation \( \frac{dy}{dx} = \frac{y+1}{x-1} \), when \( y(1) = 2 \), is
(a) zero
(b) one
(c) two
(d) infinite
Answer: (b) one

Question. The general solution of the differential equation \( y\,dx - x\,dy = 0 \) is
(a) \( xy = C \)
(b) \( x = Cy^2 \)
(c) \( y = Cx \)
(d) \( y = Cx^2 \)
Answer: (c) \( y = Cx \)

Question. A particular solution of the differential equation \( x\frac{dy}{dx} + y = 0 \), when \( x = 1 \) and \( y = 1 \), is
(a) \( y = x \)
(b) \( y = e^x \)
(c) \( y = \frac{1}{x} \)
(d) \( y = \log x \)
Answer: (c) \( y = \frac{1}{x} \)

Question. The solution of \( \frac{dy}{dx} - y = 1 \), \( y(0) = 1 \) is given by
(a) \( xy = -e^x \)
(b) \( xy = -e^{-x} \)
(c) \( xy = -1 \)
(d) \( y = 2e^x - 1 \)
Answer: (d) \( y = 2e^x - 1 \)

Question. The integrating factor of the differential equation \( (1-y^2)\frac{dx}{dy} + yx = ay \) is
(a) \( \frac{1}{y^2 - 1} \)
(b) \( \frac{1}{\sqrt{y^2 - 1}} \)
(c) \( \frac{1}{1-y^2} \)
(d) \( \frac{1}{\sqrt{1-y^2}} \)
Answer: (d) \( \frac{1}{\sqrt{1-y^2}} \)

Question. The integrating factor of differential equation \( \frac{dy}{dx} + y\tan x - \sec x = 0 \) is
(a) \( \cos x \)
(b) \( \sec x \)
(c) \( e^{\cos x} \)
(d) \( e^{\sec x} \)
Answer: (b) \( \sec x \)

Question. The integrating factor of differential equation \( \frac{dy}{dx} + y = \frac{1+y}{x} \) is
(a) \( \frac{x}{e^x} \)
(b) \( \frac{e^x}{x} \)
(c) \( xe^x \)
(d) \( e^x \)
Answer: (b) \( \frac{e^x}{x} \)

Assertion-Reason Based Questions

Question. Assertion (A): The order and degree of differential equation \( y + \frac{dy}{dx} = \frac{1}{4} \int y \,dx \), respectively are 2 and 1.
Reason (R): Order of the differential equation is the order of the highest order derivative occurring in the differential equation and the degree of differential equation is the highest power of the highest order derivative.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The order of the differential equation \( \log_e \left(1 + \frac{d^2y}{dx^2}\right) = x \) is not defined.
Reason (R): Order of the differential equation is the order of the highest order derivative occurring in the differential equation.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A): The general solution of differential equation \( \frac{y}{x}\frac{dy}{dx} = \frac{1+y^2}{1+x^2} \) is \( 1+x^2 = C(1+y^2) \).
Reason (R): \( \int \frac{x}{1+x^2} \,dx = \frac{1}{2} \log (1+x^2) + C \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The number of arbitrary constants in the solution of differential equation \( \frac{d^2y}{dx^2} = 0 \) is 2.
Reason (R): The solution of a differential equation contains as many arbitrary constants as the order of differential equation.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The equation of curve passing through \( (3, 9) \) which satisfies differential equation \( \frac{dy}{dx} = x + \frac{1}{x^2} \) is \( 6xy = 3x^3 + 29x - 6 \).
Reason (R): The solution of differential equation \( \frac{dy}{dx} = \sqrt{4-y^2} \), \( y \in (-2, 2) \) is \( y = 2\sin(x+C) \).
(a) Both A and R are correct; R is the the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A): The differential equation \( 2ye^{x/y}dx + (y - 2xe^{x/y})dy = 0 \) can be solved by making the substitution \( y = vx \).
Reason (R): A homogeneous differential equation of the form \( \frac{dx}{dy} = h\left(\frac{x}{y}\right) \) can be solved by making the substitution \( x = vy \) and of the form \( \frac{dy}{dx} = h\left(\frac{y}{x}\right) \) by substitution \( y = vx \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A): The general solution of \( \frac{dy}{dx} - \frac{2}{1+x}y = (1+x)^3 \) is \( \frac{2y}{(1+x)^2} = (1+x)^2 + C \).
Reason (R): The general solution of linear differential equation is \( y \cdot (\text{IF}) = \int Q \cdot (\text{IF}) \,dx \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): Integrating factor of \( \frac{dy}{dx} + \frac{y}{2x} = 3x^2 \) is \( x^{1/2} \).
Reason (R): Integrating factor of \( \frac{dy}{dx} + Py = Q \) is \( e^{\int P \,dx} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The general solution of the differential equation \( x\frac{dy}{dx} + 2y = x^2 \) \( (x \neq 0) \) is \( y = \frac{x^2}{4} + Cx^{-2} \).
Reason (R): The number of arbitrary constants in the general solution is equal to the order of the differential equation.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A): The differential equation \( \frac{dy}{dx} - y\cos x = 5 \) is a first order linear differential equation.
Reason (R): If P and Q are functions of x only or constant, then differential equation of the form \( \frac{dy}{dx} + Py = Q \) is a first order linear differential equation.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Case Study Based Questions - I

Order: The order of a differential equation is the highest order derivative appearing in the differential equation.
Degree: The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation in derivatives for the degree to be defined.
Based on the given information, answer the following questions:

Question. Find the sum of order and degree of differential equation \( 5\frac{d^2y}{dx^2} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} \).
Answer: On squaring both sides to eliminate the fractional power, we get: \[ \left(5\frac{d^2y}{dx^2}\right)^2 = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 \] The highest order derivative is \( \frac{d^2y}{dx^2} \), so its order is 2. The highest power of the highest order derivative is 2, so its degree is 2. Therefore, the required sum of order and degree is \( 2 + 2 = 4 \).

Question. Find the order and degree of differential equation \( (y'')^2 + 5(y')^5 + \sin x = 0 \).
Answer: The highest order derivative in the equation is \( y'' \) (the second derivative), so the order is 2. The highest power of this highest order derivative is 2, so the degree is 2.

Question. Find the order and degree of differential equation \( y\frac{dy}{dx} = \frac{x}{\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^3} \).
Answer: By cross-multiplying, we can rewrite the equation as: \[ y\left(\frac{dy}{dx}\right)^2 + y\left(\frac{dy}{dx}\right)^4 = x \] The highest order derivative in this equation is \( \frac{dy}{dx} \), so the order is 1. The highest power of this derivative is 4, so the degree is 4.

Question. Find the order and degree of differential equation \( \sqrt{a+x} \frac{dy}{dx} = -x \).
Answer: The highest order derivative occurring in the equation is \( \frac{dy}{dx} \), so the order is 1. The highest power of this highest order derivative is 1, so the degree is 1.

Question. Find the order and degree of differential equation \( y''' + y^2 = e^{y'} \).
Answer: The highest order derivative in the equation is \( y''' \) (the third derivative), so the order is 3. Since the equation cannot be written as a polynomial in its derivatives due to the term \( e^{y'} \), its degree is not defined.

Case Study Based Questions - II

An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form \( \frac{dy}{dx} = F(x,y) \) is said to be homogeneous if \( F(x,y) \) is a homogeneous function of degree zero. Whereas a function \( F(x,y) \) is a homogeneous function of degree \( n \), if \( F(\lambda x, \lambda y) = \lambda^n F(x,y) \). To solve a homogeneous differential equation of the type \( \frac{dy}{dx} = F(x,y) = g\left(\frac{y}{x}\right) \), we make the substitution \( y = vx \) and then separate the variables.
Based on the above information, answer the following questions:

Question. Show that \( (x^2 - y^2)dx + 2xydy = 0 \) is a differential equation of the type \( \frac{dy}{dx} = g\left(\frac{y}{x}\right) \).
Answer: Given differential equation is: \[ (x^2 - y^2)dx + 2xydy = 0 \] This can be rewritten as: \[ 2xydy = -(x^2 - y^2)dx \Rightarrow \frac{dy}{dx} = \frac{y^2 - x^2}{2xy} \] Dividing the numerator and the denominator of the right-hand side by \( x^2 \), we obtain: \[ \frac{dy}{dx} = \frac{\left(\frac{y}{x}\right)^2 - 1}{2\left(\frac{y}{x}\right)} \] This is clearly of the form \( \frac{dy}{dx} = g\left(\frac{y}{x}\right) \).

Question. Solve the above equation to find the general solution.
Answer: Substituting \( y = vx \) and \( \frac{dy}{dx} = v + x\frac{dv}{dx} \) into the differential equation \( \frac{dy}{dx} = \frac{y^2 - x^2}{2xy} \), we get: \[ v + x\frac{dv}{dx} = \frac{v^2 - 1}{2v} \] \[ x\frac{dv}{dx} = \frac{v^2 - 1}{2v} - v = \frac{v^2 - 1 - 2v^2}{2v} = -\frac{v^2 + 1}{2v} \] Separating variables, we get: \[ \frac{2v}{v^2 + 1} \,dv = -\frac{dx}{x} \] Integrating both sides: \[ \int \frac{2v}{v^2 + 1} \,dv = -\int \frac{dx}{x} \] \[ \log(v^2 + 1) = -\log x + \log C \Rightarrow \log(v^2 + 1) = \log\left(\frac{C}{x}\right) \] \[ v^2 + 1 = \frac{C}{x} \] Substituting back \( v = \frac{y}{x} \): \[ \frac{y^2}{x^2} + 1 = \frac{C}{x} \Rightarrow \frac{x^2 + y^2}{x^2} = \frac{C}{x} \] \[ x^2 + y^2 = Cx \] This is the required general solution.

Case Study Based Questions - III

The mixing tank shown below generates saline water (a mixture of salt and water) for the cooling of a thermoelectric power plant.
The tank initially holds 20000 L of water in which 2000 kg of salt has been dissolved. Then, pure water is poured into the tank at a rate of 5000 L per minute. The mixture in the tank, which is stirred continuously, flows out at a rate of 3000 L per minute.
The quantity of salt in the tank at time \( t \) is denoted by \( Q_t \), where \( t \) is in minutes and \( Q_t \) is in kilograms.
The rate of flow of salt into the tank is measured as \( \left(\frac{dQ_t}{dt}\right)_{\text{in}} = 0 \text{ kg/min} \).
The rate of flow of salt out of the tank is measured as: \[ \left(\frac{dQ_t}{dt}\right)_{\text{out}} = \text{Rate of flow of water out of the tank} \times \frac{\text{Quantity of salt in the tank at time } t}{\text{Amount of water in the tank at time } t} \] The rate of change of quantity of salt in the tank with respect to time is given by: \[ \frac{dQ_t}{dt} = \left(\frac{dQ_t}{dt}\right)_{\text{in}} - \left(\frac{dQ_t}{dt}\right)_{\text{out}} \] Based on the given information, answer the following questions:

Question. Find the expression for the rate of change of quantity of salt in the tank with time \( t \). Show your work.
Answer: The amount of water in the tank at any time \( t \) (in minutes) is: \[ V(t) = 20000 + (5000 - 3000)t = 20000 + 2000t \text{ L} \] The concentration of salt at time \( t \) is \( \frac{Q_t}{20000 + 2000t} \text{ kg/L} \). The rate of flow of salt out of the tank is: \[ \left(\frac{dQ_t}{dt}\right)_{\text{out}} = 3000 \times \frac{Q_t}{20000 + 2000t} = \frac{3Q_t}{20 + 2t} \text{ kg/min} \] Since the incoming water is pure, \( \left(\frac{dQ_t}{dt}\right)_{\text{in}} = 0 \). Thus, the rate of change of the quantity of salt is: \[ \frac{dQ_t}{dt} = -\frac{3Q_t}{20 + 2t} \]

Question. Find the general solution of the differential equation corresponding to the rate of change of quantity of salt in the tank with time \( t \). Show your work.
Answer: Consider the differential equation: \[ \frac{dQ_t}{dt} = -\frac{3Q_t}{20 + 2t} \] Separating the variables: \[ \frac{dQ_t}{-3Q_t} = \frac{dt}{20 + 2t} \] Integrating both sides: \[ \int \frac{dQ_t}{-3Q_t} = \int \frac{dt}{20 + 2t} \] \[ -\frac{1}{3}\ln(Q_t) = \frac{1}{2}\ln(10 + t) + C \] where \( C \) is an arbitrary constant.

Question. Use the initial conditions to determine the particular solution for the differential equation obtained. Show your work.
Answer: Given the initial condition at \( t = 0 \), \( Q_t(0) = 2000 \). Substituting these values into the general solution: \[ -\frac{1}{3}\ln(2000) = \frac{1}{2}\ln(10) + C \] \[ C = -\frac{1}{3}\ln(2 \times 10^3) - \frac{1}{2}\ln(10) \] \[ C = -\frac{1}{3}[\ln 2 + 3\ln 10] - \frac{1}{2}\ln(10) = -\frac{1}{3}\ln 2 - \frac{3}{2}\ln(10) \] Substituting \( C \) back into the general solution: \[ -\frac{1}{3}\ln Q_t = \frac{1}{2}\ln(10+t) - \frac{1}{3}\ln 2 - \frac{3}{2}\ln(10) \]

Case Study Based Questions - IV

Reshma, a Mathematics teacher of Class XII, writes a differential equation \( \frac{dy}{dx} + (\sec x)y = \tan x \) on the blackboard and asks some questions which are based on above differential equation to check the preparation of the topic, then answer the following questions:

Question. What is the form of given equation?
Answer: The given equation is a first-order linear differential equation of the form \( \frac{dy}{dx} + Py = Q \), where \( P \) and \( Q \) are functions of \( x \).

Question. Write the differential equation in the form of \( \frac{dy}{dx} + Py = Q \) and find the value of \( (P^2 - Q^2) \).
Answer: On comparing the given differential equation with \( \frac{dy}{dx} + Py = Q \), we identify \( P = \sec x \) and \( Q = \tan x \). The value of \( P^2 - Q^2 \) is: \[ P^2 - Q^2 = \sec^2 x - \tan^2 x = 1 \]

Question. Find the integration factor of equation \( \frac{dy}{dx} + y\sec x = \tan x \).
Answer: The integrating factor (\(\text{IF}\)) is: \[ \text{IF} = e^{\int P \,dx} = e^{\int \sec x \,dx} = e^{\log(\sec x + \tan x)} = \sec x + \tan x \]

Question. What is the solution of \( \frac{dy}{dx} + Py = Q \)?
Answer: The general solution is given by: \[ y \cdot (\text{IF}) = \int Q \cdot (\text{IF}) \,dx + C \] Substituting \( \text{IF} = \sec x + \tan x \) and \( Q = \tan x \): \[ y(\sec x + \tan x) = \int \tan x(\sec x + \tan x) \,dx + C \] \[ y(\sec x + \tan x) = \int (\sec x \tan x + \tan^2 x) \,dx + C \] \[ y(\sec x + \tan x) = \int \sec x \tan x \,dx + \int (\sec^2 x - 1) \,dx + C \] \[ y(\sec x + \tan x) = \sec x + \tan x - x + C \]