# Differential Calculus and Differential Equation 2016 – BSc Computer Science Part 1

#### Paper code: 13502 1502 B.sc. (Computer Science) (Part 1) Examination, 2016 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) If $y=\sin mx +\cos mx$, prove that:

$y_{n}=m^{n} \left [ 1+(-1)^{n} \sin 2mx \right ]^{\frac{1}{2}}$

(b) 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 $x^2 y_{n+2}+(2n+1)xy_{n+1}+(n^{2}+1) y_n=0$.

2. (a) State and prove Maclaurin’s theorem.

(b) Expend $\tan^{-1}x$ in powers of $\left ( x-\frac{\pi }{4} \right )$.

3. (a) If:

$y=x_{n}\log x$

prove that:

$xy_{n+1}=n!$

(b) Evaluate:

$\lim_{x \to 0}\frac{\sin x-x+\frac{x^{3}}{6}}{x^{5}}$

4. For the cardoid $r=a\left ( 1-\cos \theta \right )$, prove that:

1. $\phi =\frac {\theta}{2}$
2. $p=2a\sin^{3}\frac{\theta }{2}$
3. The Pedal equation is $2ap^{2}=r^{3}$
4. The Polar sub tangent $= 2a\sin^{2} \frac {\theta}{2} \tan \frac {\theta}{2}$

Section-B

5. (a) Solve:

$\frac {dy}{dx}=\left ( 4x+y+1\right )^{2}$

(b) Solve it:

$x\frac {dy}{dx}=y-x\tan \frac {y}{x}$

6. (a) Solve:

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

Given $y=2$ when $x=\frac {\pi}{2}$.

(b) Solve:

$xdx+ydy+\frac {xdy-ydx}{x^{2}+y^{2}}=0$

7. (a) Solve:

$\left (D^{3}+6D^{2}+11D+6 \right )y=0$

(b) Solve:

$\left (D^{2}+a^{2} \right )y=\sin ax$

8. (a) Solve:

$\left (x^{2} D^{2} + 3xD +1 \right )y=\frac {1}{\left (1-x \right )^{2}}$

Section-C

9. (a) Evaluate:

$\int \frac {5x-2}{1+2x+3x^{2}}dx$

(b) Evaluate:

$\int \sqrt{2-3x-4x^{2}}dx$

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

(b) Evaluate $\int_{0}^{\frac {\pi}{2}}\log \sin x dx$.

……End……

#### Lokesh Kumar

Being EASTER SCIENCE's founder, Lokesh Kumar wants to share his knowledge and ideas. His motive is "We assist you to choose the best", He believes in different thinking.

This site uses Akismet to reduce spam. Learn how your comment data is processed.