# IIT JAM Mathematics Test Series 2023 : Differential Equations – Their formation and solutions

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## IIT JAM Mathematics Test Series 2023 : Differential Equations – Their formation and solutions

Q1. The differential equation of the family of circles of radius ‘r’ whose centre lie on the x-axis, is

(a) y (dy/dx) + y2 = r2
(b) y {(dy/dx) + 1} = r2
(c) y2 {(dy/dx) + 1} = r2
(d) y2 {(dy/dx)2 + 1} = r2

Q2. The equation of the curve, for which the angle between the tangent and the radius vector is twice the vectorial angle is r2 ! A sin 24. This satisfies the differential equation

(a) r (dr/) = tan2θ
(b) r (/dr) = tan2θ
(c) r (dr/) = co2θ
(d) r (/dr) = cos2θ

Q3. The maximum number of linearly independent solutions of the differential equation d4y/dx4 = 0 with the condition y (0) = 1 is

(a) 4
(b) 3
(c) 2
(d) 1

Q4. The order and degree of differential equation {1 + (dy/dx)2}3/2 = k (d2y/dx2) is

(a) 3, 1
(b) 3, 2
(c) 3, 3
(d) 2, 2

Q5. Consider the following differential equations:

A. x2 (d2y/dx2) + y-2/3 {1 + (d3y/dx3)5}1/2 + d2/dx2 {(d2y/dx2)-2/3}
B. dy/dx – 6x = {ay + bx (dy/dx)}–3/2, b ≠ 0.

The sum of the order of the first differential equation and degree of the second differential equation is

(a) 6
(b) 7
(c) 8
(d) 9

Q6. The degree of the equation (d3y/dx3)2/3 + (d3y/dx3)3/2 = 0 is

(a) 3
(b) 5
(c) 4
(d) 9

Q7. Linear combinations of solutions of an ordinary differential equation are solutions if the differential equation is

(a) Linear non-homogeneous
(b) Linear homogeneous
(c) Non-linear homogeneous
(d) Non-linear non-homogeneous

Q8. Which of the following pair of functions is not a linearly independent solutions of y” + 9y = 0?

(a) sin 3x, sin 3x – cos 3x
(b) sin 3x + cos 3x, 3 sin x – 4 sin3 x
(c) sin 3x, sin 3x cos 3x
(d) sin 3x + cos 3x, 4 cos3 x – 3 cos x

Q9. Let y = Φ (x) and y = ψ(x) be solutions of y” + – 2xy’ + (sin x2) y = 0, such that Φ (0) = 1, Φ’ (0) = 1 and ψ(0) = 1, ψ'(0) =2 . The value of Wromhian W (Φ,ψ) at x = 0 is

(a) 0
(b) 1
(c) e
(d) e2

Q10. What is the degree of the differential equation for a given curve in which (subtangent)m = (subnormal)n in cartesian form, where 0 < n < m, m,n, m/n are integers?

(a) m + n
(b) m – n
(c) m*n
(d) m/n

Q11. What is the differential equation of the family of curves y = Ae3x + Be5x ; for different values of A and B.

(a) y”– 8y’ + 15y = 0
(b) y’ – 8y” + 15y = 0
(c) 8y”– y’ + 15y = 0
(d) y”– y’ + 8y = 0

Q12. Which one of the following statement is correct ? The differential equation (dy/dx)2 + 5y1/3 = x is

(a) linear equation of order 2 and degree 1
(b) nonlinear equation of order 1 and degree 2
(c) non-linear equation of order 1 and degree 6
(d) linear equation of order 1 and degree 6.

Q13. What are the order and degree respectivels of the differential equation of the family of curves y2 = 2c (x + √c) ,

(a) 1, 1
(b) 1, 2
(c) 1, 3
(d) 2, 1

Q14. Which one of the following equations has the same order and degree?

(a) d4y/dx4 + 8 (dy/dx)4 + 5x = ex
(b) 5 (d3y/dx3 )4 + 8 (dy/dx + 1)2 + 5x = x3
(c) {1 + (d3y/dx3 )}2/3 = 4 (d3y/dx3)
(d) y = x2 (dy/dx) + {(dy/dx)2 + 1}1/2

Q15. Let y1 and y2 be any two solutions of a second order linear non-homogeneous ordinary differential equation and c be any arbitrary constant. Then, in general

(a) y1 + y2 is its solution, but cy1 is not
(b) cy1 is its solution, but y1 + y2 is not
(c) both y1 + y2 and cy1 are its solutions
(d) neither y1 + y2 nor cy1 is its solution

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