Saturday, August 1, 2020

DIFFERENTIAL CALCULAS AND DIFFERENTIAL EQUATION - 2017 | B.SC CS 1ST YEAR | MJPRU | EXAM PAPER | My CS Tutorial

My CS Tutorial August 01, 2020

Differential calculas and Differential equation | My CS Tutorial

Paper Code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2017
Paper No. 1.2
DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION

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) Find nth differential coefiicient of $\tan^{-1}\frac{x}{a}$ .

(b) If $y=(x^{2}-1)^{n}$ and $P_{n}(x)=\frac{d^{n}}{dx^{n}}(x^{2}-1)^{n}$ , show that :
$\frac{d}{dx}\left \{ (1-x^{2})\frac{dP_{n}(x)}{dx} \right \}+n(n+1)P_{n}(x)=0$

2. (a) If $e^{x^{y}} = a_{0} + a_{1}x + a_{2}x^{2} +.....+ a_{n}x^{n} +.....$ , prove that :
$a_{n+1} = \frac{1}{n+1}\left \{ a_{n}+\frac{a_{n-1}}{1!} + \frac{a_{n-2}}{2!} +.....+ \frac{a_{0}}{n!} \right \}$

(b) State and prove Taylor’s theorem.

3. (a) Evaluate :

$\lim_{x\rightarrow 0}\frac{x^{1/2}\tan x}{(e^{x}-1)^{3/2}}$

(b) Show that the Pedal equation of the ellipse $\frac{x^2}{a^{2}}+\frac{y^2}{b^{2}}=1$ is:

$\frac{1}{p^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}-\frac{*r^{2}}{a^{2}b^{2}}$

4. (a) Show that the locus of the extremily of the polar subnormal of the curve $r=f(\theta )$ is $r=f'\left ( \theta - \frac{\pi}{2} \right )$ .
(b) Find the polar sub-tangent of the ellipse :

$\frac{1}{r} = 1 + e\cos \theta$

Section-B

5. (a) Solve the following :

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

(b) Solve the following :

$\frac{dy}{dx} = \frac{x-y+3}{2x-2y-5}$

6. (a) Solve the following :

$\frac{dy}{dx}+(2x + \tan^{-1}y-x^{3})(1+y^{2})=0$

(b) Solve the following :

$\frac{d^{4}y}{dx^{4}}+m^{2}y=0$

7. (a) Solve the following :

$(D^2-4D+4)y=8x^2 e^2x \sin 2x$

(b) Solve the following :

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

Section-C

8. (a) Solve:

$(3x+2)^{2}\frac{d^{2}y}{dx^{2}}+3(3x+2)\frac{dy}{dx}-36y=3x^{2}+4x+1$

(b) Evaluate the following :

$\int (3x^{2}+4x+5)^5(3x+2)dx$

9. (a) Evaluate the following :

$\int_{1}^{2}\frac{x^{3}}{(x+1)(x^{2}-7x+12)}dx$

(b) Evaluate $\int_{a}^{b}\sin x dx$ as the limit of a sum.

10. (a) Find the indefinite integral :

$\int \left \{ \sin^{2} x + \cos^{2}x + \frac{x^{3}+2x}{x^{1/2}}\right \}dx$

(b) Evaluate the following :

$\int_{0}^{\pi/4} \sec x \sqrt{\frac{1-\sin x}{1+\sin x}}dx$

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