# Differential Calculus and Differential Equation 2018 – BSc Computer Science Part 1

#### Paper code: 135021502B.sc. (Computer Science) (Part 1)Examination, 2018Paper No. 1.2DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION

##### [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 :

$\sin^{5}x \cos^{3}x$

(b) If $y=e^{a \sin^{-1} x}$, find the values of $(y_{n})_{0}$.

2. (a) Expand $2x^{3}+7x^{2}+x-1$ in power of (x-2).

(b) State and prove Maclaurin’s theorem.

3. (a) Evaluate :

$\lim_{x\rightarrow 0}\left ( \frac{\tan x}{x} \right )^{1/x^{2}}$

(b) Find the angle between the radius vector and tangent for the curve :

$r=a(1+\cos \Theta)$ at point $(r, \Theta)$

4. (a) Find the length of polar subtangent of the parabola :

$\frac{2a}{r} = 1 + \cos \Theta$

(b) Evaluate :

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

Section-B

5. (a) Evaluate :

$\int_{0}^{\pi/2} \log\sin x dx$

(b) Evaluate $\int_{a}^{b} x^{2} dx$ by summation.

6. (a) Evaluate :

$\int \frac{2x^{2}+3x+4}{x^{2}+6x+10}dx$

(b) Evaluate :

$\int \frac{1-4x-2x^{2}}{\sqrt{2x-x^{2}}}dx$

7. (a) Solve :

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

(b) Solve the following :

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

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 :

$\frac{d^{4}y}{dx^{4}}+\frac{d^{2}}{dx^{2}}+y = ax^{2} + be^{-x} \sin 2x$

9. (a) Solve :

$\frac{d^{2}y}{dx^{2}}+a^{2}y=\tan ax$

(b) Solve :

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

10. (a) Solve :

$\frac{dx}{dt} + wy =0$

$\frac{dy}{dt} - wx =0$

(b) Solve :

$\frac{d^{2}y}{dx^{2}}+a^{2}y=\cos ax$

……End……

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