Practice Questions on Formation and Solution of Differential Equations

**Questions on Formation and Solution of Differential Equations**

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

(a) y (^{dy}/_{dx}) + y^{2} = r^{2}

(b) y {(^{dy}/_{dx}) + 1} = r^{2}

(c) y^{2} {(^{dy}/_{dx}) + 1} = r^{2}

(d) y^{2} {(^{dy}/_{dx})^{2} + 1} = r^{2}

**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}/_{dθ}) = tan2θ

(b) r (^{dθ}/_{dr}) = tan2θ

(c) r (^{dr}/_{dθ}) = co2θ

(d) r (^{dθ}/_{dr}) = cos2θ

**Q3. The maximum number of linearly independent solutions of the differential equation d ^{4}y/_{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 (d^{2}y/_{dx2}) is**

(a) 3, 1

(b) 3, 2

(c) 3, 3

(d) 2, 2

**Q5. Consider the following differential equations:**

**A. x ^{2} (d^{2}y/_{dx2}) + y^{-2/3} {1 + (d^{3}y/_{dx3})^{5}}^{1/2} + d^{2}/_{dx2} {(d^{2}y/_{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 (d ^{3}y/_{dx3})^{2/3} + (d^{3}y/_{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 sin^{3} x

(c) sin 3x, sin 3x cos 3x

(d) sin 3x + cos 3x, 4 cos^{3} x – 3 cos x

**Q9. Let y = Φ (x) and y = ψ(x) be solutions of y” + – 2xy’ + (sin x ^{2}) 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) e^{2}

**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 = Ae ^{3x} + Be^{5x} ; 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} + 5y^{1/3} = 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 y ^{2} = 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) d^{4}y/_{dx4} + 8 (^{dy}/_{dx})^{4} + 5x = e^{x}

(b) 5 (d^{3}y/_{dx3} )^{4} + 8 (^{dy}/_{dx} + 1)^{2} + 5x = x^{3}

(c) {1 + (d^{3}y/_{dx3} )}^{2/3} = 4 (d^{3}y/_{dx3})

(d) y = x^{2} (^{dy}/_{dx}) + {(^{dy}/_{dx})^{2} + 1}^{1/2}

**Q15. Let y _{1} and y_{2} be any two solutions of a second order linear non-homogeneous ordinary differential equation and c be any arbitrary constant. Then, in general**

(a) y_{1} + y_{2} is its solution, but cy_{1} is not

(b) cy_{1} is its solution, but y_{1} + y_{2} is not

(c) both y_{1} + y_{2} and cy_{1} are its solutions

(d) neither y_{1} + y_{2} nor cy_{1} is its solution

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*Answer Key*

*Answer Key*

01. | (d) | 06. | (d) | 11. | (a) |

02. | (b) | 07. | (b) | 12. | (b) |

03. | (a) | 08. | (c) | 13. | (c) |

04. | (d) | 09. | (b) | 14. | (c) |

05. | (d) | 10. | (a) | 15. | (d) |