# Differential Calculus and Differential Equation 2014 – BSc Computer Science Part 1

#### Paper code: 13502 1502 B.sc. (Computer Science) (Part 1) Examination, 2014 Paper No. 1.2 DIFFERENTIAL CALCULUS AND  DIFFERENTIAL EQUATION

##### Time: Three Hours] [Maximum Marks: 50

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

Section-A

1. (a) If $I_n=\frac{d^n}{{dx}^n}(x^n \log x)$, prove that:

$I_n=nI_{n-2}+(n-1)!$

(b) If $y=tan^{-1}x$, find

$(y_n )_0$

2. (a) Apply Maclaurin’s theorem to find the expansion in ascending powers of x of $\log_{e}(1+e^x)$ to the term containing $x^{4}$.
(b) Expend $\sin x$ in powers of $\left (x-\frac{1}{2}\pi \right)$.

Section-B

3. (a) Find the equation of the normal at the point ‘t’ on the curve:

$x=a(\sin t)^{3}, x=b(\cos t)^{3}$

(b) Find the length of the polar tangent and polar normal for the curve $r=a(1+\cos\theta )$.

4. (a) Find the equation of the tangent at the point $(x_1,y_1)$ to the ellipse:

$(\frac{x^2}{a^2} +\frac{y^2}{b^2}=1)$

(b) If $r^m=a^m\cos m\theta$, prove that:

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

Section-C

5. (a) Evaluate:

$\lim_{x \to 0}\frac{\log (1-x^{2})}{\log \cos x}$

(b) Evaluate:

$\int \frac{4x-5}{2x^{2}-5x+2}dx$

6. (a) Show that:

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

(b) Evaluate:

$\lim_{x \to 0}\left [\frac{1}{n+1}+\frac{1}{n+2}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot + \frac{1}{2n} \right ]$

Section-D

7. (a) Solve:

$(xy+x)dy=(yx+y)dx$

(b) Solve:

$\cos (x+y)dy=dx$

8. (a) Solve:

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

(b) Solve:

$\sec xdy = (y+\sin x )dx$

Section-E

9. (a) Solve:

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

(b) Solve:

$(D^{2}-4D+3)y=e^{2x}\sin3x$

10. (a) Solve:

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

(b) Solve:

$\frac{dx}{dt}+4x+3y=t$

……………End……………

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