JEE Mathematics Differential Equations MCQs Set 08

Multiple Choice Questions (MCQs) for JEE Mathematics: 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.

Practice Differential Equations MCQs for JEE Mathematics

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 order of the differential equation of which \( y^2=4ax \) is a solution is:
(a) 2
(b) 1
(c) 3
(d) cannot be determined
Show Answer & Explanation

Answer: (b) 1

Explanation:
1. The relation contains one arbitrary constant a.
2. Differentiating once and eliminating a gives \( y_1=\frac{y}{2x} \).
3. The equation has order 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 2: The differential equation of all parabolas whose axis of symmetry is parallel to the y-axis has order:
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (c) 3

Explanation:
1. Such parabolas have the form \( (x-h)^2=4a(y-k) \).
2. It has three arbitrary constants h, k and a.
3. So the order is 3.

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 3: The order of the differential equation whose general solution is \( (x-a)^2+(y-b)^2=a^2 \) is:
(a) 4
(b) 3
(c) 2
(d) 1
Show Answer & Explanation

Answer: (c) 2

Explanation:
1. There are two arbitrary constants, a and b.
2. So the order of the equation 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 4: The order and degree of the differential equation of all rectangular hyperbolas whose asymptotes are the coordinate axes are respectively:
(a) 2, 2
(b) 3, 2
(c) 1, 2
(d) 1, 1
Show Answer & Explanation

Answer: (d) 1, 1

Explanation:
1. The family is \( xy=c^2 \) with one constant.
2. Differentiating once: \( x\frac{dy}{dx}+y=0 \).
3. Order 1 and 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 5: The order of the differential equation whose solution is \( y=C_1e^{x+c_2}+C_3\sin(x+c_4) \) is:
(a) 4
(b) 2
(c) 3
(d) 1
Show Answer & Explanation

Answer: (c) 3

Explanation:
1. Write \( C_1e^{x+c_2}=Ae^{x} \), which is one constant.
2. Also \( C_3\sin(x+c_4)=B\sin x+C\cos x \), which is two independent constants.
3. In all there are three independent constants, so the order is 3.

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 6: The order of the differential equation whose solution is \( y=c_1\cos(2x+c_2)-(c_3c_4)a^{x+c_5}+c_6\sin(x-c_7) \) is:
(a) 4
(b) 3
(c) 7
(d) 5
Show Answer & Explanation

Answer: (d) 5

Explanation:
1. The term \( c_1\cos(2x+c_2) \) has two independent constants.
2. \( c_3c_4a^{x+c_5}=k\,a^x \) has one.
3. \( c_6\sin(x-c_7) \) has two.
4. The total is 5, so the order is 5.

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 7: The differential equation \( \frac{dy}{dx}=\frac{4x+2y+1}{x-2y+3} \) is of the type:
(a) variable separable
(b) exact
(c) homogeneous
(d) non-homogeneous
Show Answer & Explanation

Answer: (d) non-homogeneous

Explanation:
1. Both numerator and denominator are linear in x and y with constant terms.
2. The lines \( 4x+2y+1=0 \) and \( x-2y+3=0 \) are not parallel, and the equation is not exact since \( \partial M/\partial y=-2\ne\partial N/\partial x=1 \).
3. It is a non-homogeneous equation reducible to homogeneous form.

Teacher's Note:
1. Recall the standard result for the type of equation and match it with the given form.
2. Eliminating the constants or integrating directly confirms it.

Question 8: The type of the differential equation \( x\frac{dy}{dx}+y=y^2\log x \) is:
(a) linear
(b) Bernoulli
(c) exact
(d) homogeneous
Show Answer & Explanation

Answer: (b) Bernoulli

Explanation:
1. Divide by x: \( \frac{dy}{dx}+\frac{1}{x}y=\frac{\log x}{x}y^2 \).
2. This has the form \( \frac{dy}{dx}+Py=Qy^n \) with n = 2.
3. So it is a Bernoulli equation.

Teacher's Note:
1. Compare with the standard forms in the table of solutions.
2. A quick check is to differentiate the answer.

Question 9: The type of the differential equation \( (1+y^2)\,dx=(\tan^{-1}y-x)\,dy \) is:
(a) linear in x
(b) Bernoulli
(c) linear in y
(d) homogeneous
Show Answer & Explanation

Answer: (a) linear in x

Explanation:
1. Rewrite as \( \frac{dx}{dy}+\frac{x}{1+y^2}=\frac{\tan^{-1}y}{1+y^2} \).
2. It has the form \( \frac{dx}{dy}+Px=Q \) with P and Q functions of y.
3. So it is linear in x.

Teacher's Note:
1. Recall the standard result for the type of equation and match it with the given form.
2. Eliminating the constants or integrating directly confirms it.

Question 10: The solution of \( x\,dy=\left(\frac{f(xy)}{x f'(xy)}-y\right)dx \) is:
(a) \( f(xy)=c \)
(b) \( x f(xy)=c \)
(c) \( y f(xy)=c \)
(d) \( cx=f(xy) \)
Show Answer & Explanation

Answer: (d) \( cx=f(xy) \)

Explanation:
1. \( x\,dy+y\,dx=\frac{f(xy)}{xf'(xy)}dx \) gives \( d(xy)=\frac{f(xy)}{xf'(xy)}dx \).
2. Separating, \( \int\frac{f'(xy)}{f(xy)}d(xy)=\int\frac{dx}{x} \).
3. So \( \log f(xy)=\log x+\log c \), i.e. \( f(xy)=cx \).

Teacher's Note:
1. Compare with the standard forms in the table of solutions.
2. A quick check is to differentiate the answer.

Question 11: The differential equation of all straight lines which are at a constant distance p from the origin is:
(a) \( (y-xy_1)^2=p^2(1+y_1^2) \)
(b) \( (y+xy_1)^2=p^2(1-y_1^2) \)
(c) \( (y+xy_1)^2=p^2(1+y_1^2) \)
(d) \( (y-xy_1)^2=p^2(1-y_1^2) \)
Show Answer & Explanation

Answer: (a) \( (y-xy_1)^2=p^2(1+y_1^2) \)

Explanation:
1. Take the line \( x\cos\alpha+y\sin\alpha=p \).
2. Differentiating, \( \cos\alpha+y_1\sin\alpha=0 \), so \( \cos\alpha=-y_1\sin\alpha \).
3. Substituting, \( \sin\alpha\,(y-xy_1)=p \).
4. Squaring and using \( \sin^2\alpha=\frac{1}{1+y_1^2} \) gives \( (y-xy_1)^2=p^2(1+y_1^2) \).

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 12: If f, g, h are continuous, thrice differentiable functions each satisfying \( \frac{d^2y}{dx^2}+y=0 \) and \( \phi(x)=\begin{vmatrix}f&g&h\\f_1&g_1&h_1\\f_2&g_2&h_2\end{vmatrix} \) (subscript k denotes the k-th derivative), then \( \phi(x) \) equals:
(a) 0
(b) f(x)g(x)h(x)
(c) f_1(x)g_1(x)h_1(x)
(d) 1
Show Answer & Explanation

Answer: (a) 0

Explanation:
1. Each function satisfies \( f_2=-f,\ g_2=-g,\ h_2=-h \).
2. So the third row is the negative of the first row.
3. Two proportional rows make the determinant 0.

Teacher's Note:
1. Compare with the standard forms in the table of solutions.
2. A quick check is to differentiate the answer.

Question 13: If \( \phi'(x)=\phi(x) \) and \( \phi(1)=2 \), then \( \phi(3) \) equals:
(a) \( e^2 \)
(b) \( e^3 \)
(c) \( 3e^2 \)
(d) \( 2e^2 \)
Show Answer & Explanation

Answer: (d) \( 2e^2 \)

Explanation:
1. \( \frac{\phi'}{\phi}=1 \) gives \( \phi(x)=Ce^x \).
2. \( \phi(1)=Ce=2 \), so \( C=2/e \).
3. \( \phi(3)=\frac{2}{e}e^3=2e^2 \).

Teacher's Note:
1. Recall the standard result for the type of equation and match it with the given form.
2. Eliminating the constants or integrating directly confirms it.

Question 14: The differential equation obtained by eliminating A and B from \( y=A\cos\omega t+B\sin\omega t \) is:
(a) \( y''+y'=0 \)
(b) \( y''-\omega^2y=0 \)
(c) \( y''+\omega^2y=0 \)
(d) \( y''+y=0 \)
Show Answer & Explanation

Answer: (c) \( y''+\omega^2y=0 \)

Explanation:
1. Differentiate twice with respect to t.
2. \( y''=-\omega^2(A\cos\omega t+B\sin\omega t)=-\omega^2y \).
3. So \( y''+\omega^2y=0 \).

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

Answer: (c) 2

Explanation:
1. The highest derivative present is \( \frac{d^2y}{dx^2} \).
2. 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 16: The order and degree of the differential equation \( \frac{dy}{dx}+2\frac{dx}{dy}=7 \) are:
(a) 1, 1
(b) 1, 2
(c) 2, 1
(d) 2, 2
Show Answer & Explanation

Answer: (b) 1, 2

Explanation:
1. Since \( \frac{dx}{dy}=\frac{1}{y_1} \), the equation is \( y_1+\frac{2}{y_1}=7 \).
2. Multiply by \( y_1 \): \( y_1^2-7y_1+2=0 \).
3. Order 1, degree 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 17: The degree of the differential equation \( x=1+\frac{dy}{dx}+\frac{1}{2!}\left(\frac{dy}{dx}\right)^2+\frac{1}{3!}\left(\frac{dy}{dx}\right)^3+\cdots \) is:
(a) three
(b) one
(c) not defined
(d) four
Show Answer & Explanation

Answer: (b) one

Explanation:
1. The series on the right is the expansion of \( e^{dy/dx} \).
2. So \( x=e^{dy/dx} \), which gives \( \frac{dy}{dx}=\log x \).
3. The degree is one.

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 18: The degree of the differential equation of all circles in the first quadrant touching the coordinate axes is:
(a) 2
(b) 1
(c) 3
(d) 4
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. Such circles are \( (x-r)^2+(y-r)^2=r^2 \) with one parameter r.
2. Differentiating, \( (x-r)+(y-r)y_1=0 \), so \( r=\frac{x+yy_1}{1+y_1} \).
3. Eliminating r gives \( (x-y)^2(1+y_1^2)=(x+yy_1)^2 \).
4. The highest power of \( y_1 \) is 2, so the degree 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 19: The order of the differential equation satisfied by \( \sqrt{1-x^4}+\sqrt{1-y^4}=a(x^2-y^2) \) is:
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (a) 1

Explanation:
1. The relation contains one arbitrary constant a.
2. So the order of the equation is 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 20: The differential equation representing the family of curves \( y^2=2c(x+\sqrt c) \), where c is a parameter, is of:
(a) degree 4
(b) order 2
(c) degree 3
(d) degree 1
Show Answer & Explanation

Answer: (c) degree 3

Explanation:
1. Differentiating, \( yy_1=c \).
2. Substituting: \( y^2=2yy_1x+2(yy_1)^{3/2} \).
3. Isolate the radical and square: \( (y^2-2xyy_1)^2=4y^3y_1^3 \).
4. The equation has order 1 and degree 3.

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 21: The order of the differential equation of all conics whose centre lies at the origin is:
(a) 2
(b) 3
(c) 4
(d) 1
Show Answer & Explanation

Answer: (b) 3

Explanation:
1. Such conics are \( ax^2+2hxy+by^2=1 \).
2. They contain three independent constants a, h, b.
3. So the order is 3.

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 22: The degree of the differential equation of which \( y^2=4a(x+a) \) is a solution is:
(a) 2
(b) 3
(c) 1
(d) 4
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. Differentiating, \( yy_1=2a \).
2. Substituting: \( y^2=2yy_1x+y^2y_1^2 \), i.e. \( y=2xy_1+yy_1^2 \).
3. The degree 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.

Free study material for Differential Equations

Practice MCQs for JEE Mathematics Differential Equations

JEE Mathematics Differential Equations Objective Test Questions

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

NCERT-Aligned Objective Questions and Solutions

Built using the official NCERT book for JEE, these Mathematics objective sets provide reliable academic guidance. Pair your practice with our recommended NCERT solutions to master optimal problem-solving approaches.

Next Steps in Your Exam Preparation

Explore our broader library of printable assignments, chapter notes, and mock tests designed to support continuous revision and secure higher marks in JEE assessments.

FAQs

Where can I access latest JEE Mathematics Differential Equations MCQs Set 08?

You can get most exhaustive JEE Mathematics Differential Equations MCQs Set 08 for free on StudiesToday.com. These MCQs for JEE Mathematics are updated for the 2026-27 academic session as per JEE examination standards.

Are Assertion-Reasoning and Case-Study MCQs included in the Mathematics JEE material?

Yes, our JEE Mathematics Differential Equations MCQs Set 08 include the latest type of questions, such as Assertion-Reasoning and Case-based MCQs. 50% of the JEE paper is now competency-based.

How do practicing Mathematics MCQs help in scoring full marks in JEE exams?

By solving our JEE Mathematics Differential Equations MCQs Set 08, JEE students can improve their accuracy and speed which is important as objective questions provide a chance to secure 100% marks in the Mathematics.

Do you provide answers and explanations for JEE Mathematics Differential Equations MCQs Set 08?

Yes, Mathematics MCQs for JEE have answer key and brief explanations to help students understand logic behind the correct option as its important for 2026 competency-focused JEE exams.

Can I practice these Mathematics JEE MCQs online?

Yes, you can also access online interactive tests for JEE Mathematics Differential Equations MCQs Set 08 on StudiesToday.com as they provide instant answers and score to help you track your progress in Mathematics.