Differential Calculus and Differential Equations 2022 – BSc Computer Science Part 1

July 9, 2023July 9, 2023 Lokesh Kumar 0 Comments 1st Year, BSc, BSc Computer Science, Computer Science, Exam paper

paper.html

Paper code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2022
Paper No. 1.2
Differential Calculus and Differential Equations

Time: Three Hours]

[Maximum Marks: 50

Note: Attempt five questions in all selecting one question from each Section. All questions carry equal marks.

Section-A

1. (a) If $y=a\cos\left ( \log x \right )+b\sin\left ( \log x \right )$ show that $x^{2}y_{2}+xy_{1}+y=0$ and $x^{2}y_{n+2}+\left ( 2n+1 \right )xy_{n+1}+\left ( n^{2}+1 \right )y_{n}=0$ .

(b) Find $(y_{n})_{0}$ when $y = \sin(a \sin^{-1}x$ .

2. (a) Expand log(1+x) with the help of Maclavrin’s theorem.

(b) Expand sin x in powers of (x-?/2) by Taylor’s theorem.

3. (a) Find the sub-tangent and subnormal at the point t on the cycloid.

$x=a(t+\sin t), y = a(1-\cos t)$

(b) Find the pedal equation of the parabola :

y² = 4a (x + a)

4. (a) In the curve r^m = a^m cos m? prove that :

$\frac{ds}{d\theta}=a \sec^{\frac{m-1}{m}}n\theta$

(b) Find the following limit :

$\lim_{x \to 0}\frac{\sin x -x + \frac{x^{3}}{6}}{x^{2}}$

Section-B

5. (a) Find :

$\int \frac{dx}{x^{3}-1}$

(b) Find :

$\int \frac{x}{\left ( x-3 \right )\sqrt{x+1}}dx$

6. (a) Show that :

$\int_{0}^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+ \sqrt{\cos x}}dx = \frac{1}{4}\pi$

(b) From the definition of a definite integral as the limit of a sum evaluate:

$\int_{a}^{b}e^{x}dx$

7. (a) Solve the differential equation :

$\frac{dy}{dx} = e^{x-y} + x^{2}e^{-y}$

(b) Solve the differential equation :

$\frac{dy}{dx} + \frac{y}{x} = x^{2}$

Section-C

8. Solve the following differential equations :

(a)

$\frac{d^{2}y}{dx^{2}}+ \frac{dy}{dx} + y = e^{x}$

(b)

$\frac{d^{3}y}{dx^{3}}+ y = \cos 2x$

9. Solve the following differential equations :

(a)

$x^{2}\frac{d^{2}y}{dx^{2}}+2x\frac{dy}{dx}-20y = (x+1)^{2}$

(b)

$x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+4y = 2x^{2}$

10. (a) Solve:

$[latex]\frac{dx}{dt}-y=t, \frac{dy}{dt}+x=1$ [/latex]

(b) Solve:

$[latex]\frac{dx}{dt}+7x-y=0, \frac{dy}{dt}+2x+jy=0$ [/latex]

……End……

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