JEE Mathematics Differential Equations MCQs Set 10

Mathematics Objective Questions and Answers: Differential Equations

Review structured MCQ sets for JEE Mathematics Differential Equations. Built according to official JEE guidelines, these downloadable questions support daily revision and core concept reinforcement.

Download Differential Equations MCQs with Answers

Navigate directly to the 50 objective questions for Differential Equations using the digital viewer below. Each practice set includes verified answer keys, allowing students to instantly cross-check their work and identify areas requiring further revision.

Multiple Choice Questions

Question 1: The substitution required to change \( (3y-7x+7)dx+(7y-3x+3)dy=0 \) to a homogeneous differential equation is:
(a) \( x=X+1,\ y=Y \)
(b) \( x=X,\ y=Y+1 \)
(c) \( x=X+1,\ y=Y+1 \)
(d) \( x=X+2,\ y=Y+2 \)
Show Answer & Explanation

Answer: (a) \( x=X+1,\ y=Y \)

Explanation:
1. Solve \( 3y-7x+7=0 \) and \( 7y-3x+3=0 \) together.
2. From the first, \( y=\frac{7x-7}{3} \); substituting, \( 49x-49-9x+9=0 \), so \( x=1,\ y=0 \).
3. Shift the origin to (1, 0).

Teacher's Note:
1. For (ax+by+c)/(a'x+b'y+c') find the intersection of the lines first.
2. If the lines are parallel use v = ax + by.

Question 2: The integrating factor of \( (1-y^2)\frac{dx}{dy}+yx=ay \) (\( -1(a) \( \frac{1}{y^2-1} \)
(b) \( \frac{1}{\sqrt{y^2-1}} \)
(c) \( \frac{1}{1-y^2} \)
(d) \( \frac{1}{\sqrt{1-y^2}} \)
Show Answer & Explanation

Answer: (d) \( \frac{1}{\sqrt{1-y^2}} \)

Explanation:
1. Standard form: \( \frac{dx}{dy}+\frac{y}{1-y^2}x=\frac{ay}{1-y^2} \).
2. \( \int\frac{y}{1-y^2}dy=-\frac12\log(1-y^2) \).
3. IF \( =e^{-\frac12\log(1-y^2)}=\frac{1}{\sqrt{1-y^2}} \).

Teacher's Note:
1. Write the equation as dy/dx + Py = Q; the integrating factor is e raised to the integral of P dx.
2. Constants in the integral of P are not needed.

Question 3: The integrating factor of \( x\log x\frac{dy}{dx}+y=2\log x \) is:
(a) \( x \)
(b) \( \log x \)
(c) \( \log(\log x) \)
(d) \( \frac2x \)
Show Answer & Explanation

Answer: (b) \( \log x \)

Explanation:
1. Divide by \( x\log x \): \( \frac{dy}{dx}+\frac{y}{x\log x}=\frac2x \).
2. \( \int\frac{dx}{x\log x}=\log\log x \).
3. IF \( =\log x \).

Teacher's Note:
1. Bring the equation to standard linear form before reading P.
2. If the equation is linear in x, use dx/dy + Px = Q and e raised to the integral of P dy.

Question 4: The integrating factor of \( (1+x^2)\frac{dy}{dx}+y=e^{\tan^{-1}x} \) is:
(a) \( \tan^{-1}x \)
(b) \( e^{\tan^{-1}x} \)
(c) \( \frac{1}{1+x^2} \)
(d) \( \mathrm{Tan}\,x \)
Show Answer & Explanation

Answer: (b) \( e^{\tan^{-1}x} \)

Explanation:
1. Standard form: \( \frac{dy}{dx}+\frac{1}{1+x^2}y=\frac{e^{\tan^{-1}x}}{1+x^2} \).
2. \( \int\frac{dx}{1+x^2}=\tan^{-1}x \).
3. IF \( =e^{\tan^{-1}x} \).

Teacher's Note:
1. Write the equation as dy/dx + Py = Q; the integrating factor is e raised to the integral of P dx.
2. Constants in the integral of P are not needed.

Question 5: The integrating factor of \( \frac{dy}{dx}=\frac{x+y+1}{x+1} \) is:
(a) \( \frac{1}{y+1} \)
(b) \( \frac{1}{x+1} \)
(c) \( \log(x+1) \)
(d) \( \log(y+1) \)
Show Answer & Explanation

Answer: (b) \( \frac{1}{x+1} \)

Explanation:
1. \( \frac{dy}{dx}-\frac{y}{x+1}=1 \).
2. \( P=-\frac{1}{x+1} \), so \( \int P\,dx=-\log(x+1) \).
3. IF \( =\frac{1}{x+1} \).

Teacher's Note:
1. Bring the equation to standard linear form before reading P.
2. If the equation is linear in x, use dx/dy + Px = Q and e raised to the integral of P dy.

Question 6: The integrating factor of \( \frac{dy}{dx}-y\cot x=\mathrm{cosec}\,x \) is:
(a) \( \sin x \)
(b) \( \tan x \)
(c) \( \sec x \)
(d) \( \mathrm{cosec}\,x \)
Show Answer & Explanation

Answer: (d) \( \mathrm{cosec}\,x \)

Explanation:
1. \( P=-\cot x \).
2. \( \int P\,dx=-\log\sin x \).
3. IF \( =\frac{1}{\sin x}=\mathrm{cosec}\,x \).

Teacher's Note:
1. Write the equation as dy/dx + Py = Q; the integrating factor is e raised to the integral of P dx.
2. Constants in the integral of P are not needed.

Question 7: The solution of \( \frac{dy}{dx}+2y=e^{-x} \) is:
(a) \( ye^x=1+ce^{-x} \)
(b) \( ye^{-x}=1+ce^{-x} \)
(c) \( yx=e^{-x}+c \)
(d) \( e^xy=e^{-x}+ce^{-2x} \)
Show Answer & Explanation

Answer: (a) \( ye^x=1+ce^{-x} \)

Explanation:
1. IF \( =e^{2x} \).
2. \( ye^{2x}=\int e^{2x}e^{-x}dx=e^x+c \).
3. \( y=e^{-x}+ce^{-2x} \), so \( ye^x=1+ce^{-x} \).

Teacher's Note:
1. Standard form dy/dx + Py = Q gives y times IF = integral of Q times IF.
2. Do not forget the constant of integration.

Question 8: The order of \( (3x+2y)\left(\frac{dy}{dx}\right)^2+5x\frac{d^2y}{dx^2}=0 \) is:
(a) 2
(b) 1
(c) 4
(d) 3
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. The highest derivative is \( \frac{d^2y}{dx^2} \).
2. So the order is 2.

Teacher's Note:
1. Count independent arbitrary constants to find the order of the equation of a family.
2. Degree is defined only when the equation is polynomial in the derivatives.

Question 9: The order and degree of \( \sqrt{1+\left(\frac{d^2y}{dx^2}\right)^3}=\left(2+\frac{dy}{dx}\right)^{1/3} \) are:
(a) 2, 3
(b) 2, 9
(c) 2, 6
(d) 2, 2
Show Answer & Explanation

Answer: (b) 2, 9

Explanation:
1. Raise both sides to the power 6: \( \left(1+y_2^3\right)^3=(2+y_1)^2 \).
2. The highest derivative is \( y_2 \) and expanding the left side gives \( y_2^9 \).
3. Order 2, degree 9.

Teacher's Note:
1. The order is the highest derivative present; the degree is its power after the equation is made a polynomial in the derivatives.
2. Remove radicals and fractional powers before reading off the degree.

Question 10: The order and degree of \( \frac{d^2y}{dx^2}=\sin\left(\frac{dy}{dx}\right)+xy \) are respectively:
(a) 2, 1
(b) 2, infinite
(c) 2, 0
(d) 2, not defined
Show Answer & Explanation

Answer: (d) 2, not defined

Explanation:
1. The order is 2.
2. The derivative \( y_1 \) appears inside the sine, so the equation is not a polynomial in the derivatives.
3. The degree is not defined.

Teacher's Note:
1. Count independent arbitrary constants to find the order of the equation of a family.
2. Degree is defined only when the equation is polynomial in the derivatives.

Question 11: The order of the differential equation of the family of all concentric circles centred at the fixed point (h, k) is:
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (a) 1

Explanation:
1. The circles are \( (x-h)^2+(y-k)^2=r^2 \) with h, k fixed.
2. Only the radius r is arbitrary.
3. So the order is 1.

Teacher's Note:
1. The order is the highest derivative present; the degree is its power after the equation is made a polynomial in the derivatives.
2. Remove radicals and fractional powers before reading off the degree.

Question 12: The number of arbitrary constants in the solution of a differential equation of degree 3 and order 2 is:
(a) 2
(b) 3
(c) 5
(d) 4
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. The number of independent arbitrary constants in the general solution equals the order.
2. The order is 2, so there are 2 constants.

Teacher's Note:
1. Count independent arbitrary constants to find the order of the equation of a family.
2. Degree is defined only when the equation is polynomial in the derivatives.

Question 13: The degree and order of the differential equation of the family of all parabolas whose axis is the x-axis are respectively:
(a) 2, 1
(b) 1, 2
(c) 3, 2
(d) 2, 2
Show Answer & Explanation

Answer: (b) 1, 2

Explanation:
1. The family is \( y^2=4a(x+h) \) with two constants, so the order is 2.
2. Differentiating twice: \( y\,y_2+y_1^2=0 \), which is of degree 1.

Teacher's Note:
1. Count independent arbitrary constants to find the order of the equation of a family.
2. Degree is defined only when the equation is polynomial in the derivatives.

Question 14: The order of the differential equation of all conics whose axes coincide with the coordinate axes is:
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (b) 2

Explanation:
1. Such conics are \( ax^2+by^2=1 \).
2. There are two constants, so the order is 2.

Teacher's Note:
1. The order is the highest derivative present; the degree is its power after the equation is made a polynomial in the derivatives.
2. Remove radicals and fractional powers before reading off the degree.

Question 15: The family of curves \( y=Ax+A^4 \) is represented by a differential equation of degree:
(a) 3
(b) 2
(c) 4
(d) 1
Show Answer & Explanation

Answer: (c) 4

Explanation:
1. Differentiating, \( y_1=A \).
2. So \( y=xy_1+y_1^4 \).
3. The degree is 4.

Teacher's Note:
1. Count independent arbitrary constants to find the order of the equation of a family.
2. Degree is defined only when the equation is polynomial in the derivatives.

Question 16: The differential equation whose solution is \( y=a\cos(3x+b) \) is:
(a) \( y_2+3y=0 \)
(b) \( y_2+y=0 \)
(c) \( y_2+9y=0 \)
(d) \( y_2+6y=0 \)
Show Answer & Explanation

Answer: (c) \( y_2+9y=0 \)

Explanation:
1. \( y_1=-3a\sin(3x+b) \), \( y_2=-9a\cos(3x+b)=-9y \).
2. So \( y_2+9y=0 \).

Teacher's Note:
1. Differentiate as many times as there are arbitrary constants, then eliminate them.
2. The order of the resulting equation equals the number of independent constants.

Question 17: The differential equation whose solution is \( y=(c_1x+c_2)e^{5x} \) is:
(a) \( y_2+10y_1+24y=0 \)
(b) \( y_2-10y_1+25y=0 \)
(c) \( y_2-5y_1+25y=0 \)
(d) \( y_2-5y_1+10y=0 \)
Show Answer & Explanation

Answer: (b) \( y_2-10y_1+25y=0 \)

Explanation:
1. The repeated root is 5.
2. \( (m-5)^2=m^2-10m+25 \).
3. So \( y_2-10y_1+25y=0 \).

Teacher's Note:
1. Write the family with its parameters, differentiate repeatedly and eliminate the parameters.
2. Check the final answer by differentiating the given solution once more.

Question 18: The differential equation whose solution is \( y=ce^x \) is:
(a) \( y_1=y^2 \)
(b) \( y_2=y_1 \)
(c) \( y_1=y \)
(d) \( y_1+y=0 \)
Show Answer & Explanation

Answer: (c) \( y_1=y \)

Explanation:
1. Differentiating, \( y_1=ce^x=y \).
2. So \( y_1=y \).

Teacher's Note:
1. Differentiate as many times as there are arbitrary constants, then eliminate them.
2. The order of the resulting equation equals the number of independent constants.

Question 19: The differential equation whose solution is \( y=A\cos x+\sin x \) is:
(a) \( \frac{dy}{dx}+y\tan x=\sec x \)
(b) \( \frac{dy}{dx}+y\sin x=\cos x \)
(c) \( \frac{dy}{dx}+y\sec x=\tan x \)
(d) \( \frac{dy}{dx}+y\cot x=\mathrm{cosec}\,x \)
Show Answer & Explanation

Answer: (a) \( \frac{dy}{dx}+y\tan x=\sec x \)

Explanation:
1. Differentiating, \( y_1=-A\sin x+\cos x \).
2. From the solution, \( A=\frac{y-\sin x}{\cos x} \).
3. So \( y_1=-(y-\sin x)\tan x+\cos x \), i.e. \( y_1+y\tan x=\sin x\tan x+\cos x=\sec x \).

Teacher's Note:
1. Write the family with its parameters, differentiate repeatedly and eliminate the parameters.
2. Check the final answer by differentiating the given solution once more.

Question 20: The differential equation whose solution is \( y=A\sin2x+B\cos2x \) is:
(a) \( y_2=4y \)
(b) \( y_2+3y=0 \)
(c) \( y_2+y=0 \)
(d) \( y_2+4y=0 \)
Show Answer & Explanation

Answer: (d) \( y_2+4y=0 \)

Explanation:
1. \( y_2=-4(A\sin2x+B\cos2x)=-4y \).
2. So \( y_2+4y=0 \).

Teacher's Note:
1. Differentiate as many times as there are arbitrary constants, then eliminate them.
2. The order of the resulting equation equals the number of independent constants.

Question 21: The differential equation obtained by eliminating A, B, C from \( y=e^{3x}(A+Bx+Cx^2) \) is:
(a) \( y_3+9y_2-27y_1+27y=0 \)
(b) \( y_3+9y_2+27y_1+27y=0 \)
(c) \( y_3-9y_2-27y_1-27y=0 \)
(d) \( y_3-9y_2+27y_1-27y=0 \)
Show Answer & Explanation

Answer: (d) \( y_3-9y_2+27y_1-27y=0 \)

Explanation:
1. The root 3 is repeated three times.
2. \( (m-3)^3=m^3-9m^2+27m-27 \).
3. So \( y_3-9y_2+27y_1-27y=0 \).

Teacher's Note:
1. Write the family with its parameters, differentiate repeatedly and eliminate the parameters.
2. Check the final answer by differentiating the given solution once more.

Question 22: The differential equation of all non-horizontal lines in a plane is:
(a) \( \frac{d^2y}{dx^2}=0 \)
(b) \( \frac{d^2x}{dy^2}=0 \)
(c) \( \frac{dy}{dx}=0 \)
(d) \( \frac{dx}{dy}=0 \)
Show Answer & Explanation

Answer: (b) \( \frac{d^2x}{dy^2}=0 \)

Explanation:
1. A non-horizontal line can be written \( x=my+c \).
2. Differentiating twice with respect to y: \( \frac{d^2x}{dy^2}=0 \).

Teacher's Note:
1. Differentiate as many times as there are arbitrary constants, then eliminate them.
2. The order of the resulting equation equals the number of independent constants.

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Differential Equations Objective Questions & Solutions for JEE Mathematics

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