Differential Equations Questions with Solutions

14 questions1 easy10 medium3 hard

A differential equation is an equation involving a function and its derivatives. This topic covers the order and degree of a differential equation, forming a differential equation by eliminating arbitrary constants, and solving first-order equations. The main solution methods are variables separable, homogeneous equations (using the substitution y = vx), and linear equations of the form dy/dx + Py = Q, solved with the integrating factor e^(integral of P dx). Exam questions also include application problems on growth and decay, such as population growth, radioactive decay and Newton's law of cooling. Remember that the degree is defined only when the equation is a polynomial in its derivatives. Practise the differential equations questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
A company models the rate of change of concentration by $(y^{2} + 3 x y), d x + (x^{2} + x y), d y = 0$ , with $y = 1$ when $x = 1$ . Find $y$ as a function of $x$ (choose the correct option).
A. $\frac{x ^{2}}{x + 1}$ B. $\frac{x + 1}{x ^{2}}$ C. $x + 1$ D. $y = x (- 1 + 1 + \frac{3}{x ^{4}})$
mediumAsked in MAH MCA CET 2025View solution ›
Q2.
A family of parabolas $y = a x^{2}$ $(a > 0)$ has vertices on the $x$ –axis and axis parallel to $y$ –axis. The differential equation of this family is:
A. $y^{″} - \frac{2}{x ^{2}} y^{'} = 0$ B. $y^{'} = 2 a x$ C. $x y^{'} - 2 y = 0$ D. $y = y^{'}, x$
mediumAsked in MAH MCA CET 2025View solution ›
Q3.
Given $, \frac{d y}{d x} + y = 1,$ with $y (0) = 2$ , find $y (1)$ .
A. $2 e^{- 1} + 1$ B. $e^{- 1} + 2$ C. $1 + e^{- 1}$ D. $1 - e^{- 1}$
mediumAsked in MAH MCA CET 2025View solution ›
Q4.
The population of a town grows at the rate of 10% per year. How long will it take for the population to triple?
A. $5 l o g 3$ B. $10 l o g 3$ C. $\frac{l o g 3}{10}$ D. $3 l o g 10$
mediumAsked in MAH MCA CET 2025View solution ›
Q5.
A company models the rate of change of concentration by $(y^{2} + 3 x y) d x + (x^{2} + x y) d y = 0$ , with $y = 1$ when $x = 1$ . Find $y$ as a function of $x$ .
A. $\frac{x ^{2}}{x + 1}$ B. $\frac{x + 1}{x ^{2}}$ C. $x + 1$ D. $x (- 1 + 1 + \frac{3}{x ^{4}})$
hardAsked in MAH MCA CET 2025View solution ›
Q6.
A family of parabolas $y = a x^{2}$ $(a > 0)$ has vertices on the $x$ -axis and axis parallel to $y$ -axis. The differential equation of this family is:
A. $y ″ - \frac{2}{x ^{2}} y' = 0$ B. $y' = 2 a x$ C. $x y' - 2 y = 0$ D. $y = x y'$
mediumAsked in MAH MCA CET 2025View solution ›
Q7.
Given $\frac{d y}{d x} + y = 1$ with $y (0) = 2$ , find $y (1)$ .
A. $2 e^{- 1} + 1$ B. $e^{- 1} + 2$ C. $1 + e^{- 1}$ D. $1 - e^{- 1}$
easyAsked in MAH MCA CET 2025View solution ›
Q8.
A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant $= k > 0$ ). Find the time after which the cone is empty.
A. $T = \frac{H}{k}$ B. $T = H k$ C. T=H%kD. T=0
hardAsked in MAH MCA CET 2024View solution ›
Q9.
The differential equation of the family of curves $y = e^{x} (A cos x + B s in x)$ , where A and B are arbitrary constants, is
A. $\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + 2 y = 0$ B. $\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} - 2 y = 0$ C. $\frac{d ^{2} y}{d x ^{2}} + (\frac{d y}{d x})^{2} + y = 0$ D. $\frac{d ^{2} y}{d x ^{2}} - 7 \frac{d y}{d x} + 2 y = 0$
mediumAsked in MAH MCA CET 2024View solution ›
Q10.
A particle is at rest at the origin. It moves along the x-axis with an acceleration $x - x^{2}$ , where $x$ is the distance of the particle at time $t$ . The particle next comes to rest after it has covered a distance
A. 1B. $\frac{1}{2}$ C. $\frac{3}{2}$ D. None of these
mediumAsked in NIMCET 2022View solution ›
Q11.
The solution of $(e^{x} + 1) y d y = (y + 1) e^{x} d x$ is
A. $e^{y} = c (e^{x} + 1) (y + 1)$ B. $e^{y} = e^{x} + y + 1$ C. $y = c (e^{x} + 1) (y + 1)$ D. None of these
mediumAsked in NIMCET 2017View solution ›
Q12.
The solution of the differential equation $\frac{d y}{d x} = e^{x + y} + x^{2} e^{y}$ is
A. $e^{x - y} + \frac{x ^{3}}{3} = c$ B. $e^{x} + e^{- y} + \frac{x ^{3}}{3} = c$ C. $e^{x} - e^{- y} + \frac{x ^{3}}{3} = c$ D. None of these
mediumAsked in NIMCET 2017View solution ›
Q13.
Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of
A. circlesB. parabolasC. hyperbolasD. straight lines
hardAsked in NIMCET 2018View solution ›
Q14.
The curve satisfying the differential equation $y d x - (x + 3 y^{2}) d y = 0$ and passing through the point (1, 1) also passes through the point
A. $(\frac{1}{4}, \frac{1}{2})$ B. $(\frac{1}{4}, - \frac{1}{2})$ C. $(- \frac{1}{3}, \frac{1}{3})$ D. $(\frac{1}{3}, - \frac{1}{3})$
mediumAsked in NIMCET 2019View solution ›