DIFFERENTIAL EQUATIONS

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Multiple Choice questions-

(a) 3

(b) 2

(c) 1

(d) not defined.

2. The order of the differential equation:

(a) 2

(b) 1

(c) 0

(d) not defined.

3. The number of arbitrary constants in the general solution of a differential equation of fourth order is:

(a) 0

(b) 2

(c) 3

(d) 4.

4. The number of arbitrary constants in the particular solution of a differential equation of third order is:

(a) 3

(b) 2

(c) 1

(d) 0.

5. Which of the following differential equations has y = c₁ e^x + c₂ e^-x as the general solution?

(a) $\frac{d^{2}y}{d{x}^2}$ + y = 0

(b) $\frac{d^{2}y}{d{x}^2}$ – y = 0

(c) $\frac{d^{2}y}{d{x}^2}$ + 1 = 0

(d) $\frac{d^{2}y}{d{x}^2}$ – 1 = 0

6. Which of the following differential equations has y = x as one of its particular solutions?

(a) $\frac{d^{2}y}{d{x}^2}$ – x² $\frac{dy}{dx}$ + xy = x

(b) $\frac{d^{2}y}{d{x}^2}$ + x $\frac{dy}{dx}$ + xy = x

(c) $\frac{d^{2}y}{d{x}^2}$ – x² $\frac{dy}{dx}$ + xy = 0

(d) $\frac{d^{2}y}{d{x}^2}$ + x $\frac{dy}{dx}$ + xy = 0

7. The general solution of the differential equation $\frac{dy}{dx}$ = e^x+y is

(a) e^x + e^-y = c

(b) e^x + e^y = c

(c) e^-x + e^y = c

(d) e^-x + e^-y = c.

8. Which of the following differential equations cannot be solved, using variable separable method?

(a) $\frac{dy}{dx}$ + e^x+y + e^-x+y

(b) (y² – 2xy) dx = (x² – 2xy) dy

(c) xy$\frac{dy}{dx}$ = 1 + x + y + xy

(d) $\frac{dy}{dx}$ + y = 2.

9. A homogeneous differential equation of the form $\frac{dy}{dx}$ = h($\frac{x}{y}$) can be solved by making the substitution.

(a) y = vx

(b) v = yx

(c) x = vy

(d) x = v

10. Which of the following is a homogeneous differential equation?

(a) (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0

(b) xy dx – (x³ + y²)dy = Q

(c) (x³ + 2y²) dx + 2xy dy = 0

(d) y² dx + (x² – xy – y²)dy = 0.

  Very Short Questions:

Short Questions:

Long Questions:

Case Study Questions:

1. If the equation is of the form  where P, Q are functions of x, then the solution of the differential equation is given by ye is called the integrating factor (I.F.).

Based on the above information, answer the following questions.

2. If the equation is of the form  where f(x, y), g(x, y) are homogeneous functions of the same degree in x and y, then put y = vx And   so that the dependent variable y is changed to another variable v and then apply variable separable method.

Based on the above information, answer the following questions.

Answer Key-

Multiple Choice questions-

Very Short Answer:

Short Answer:

Long Answer:

Case Study Answers:

1. Answer :

2. Answer :

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