RGU Question Paper Mathematics Semester II 2020: Calculus & Differential Equations (Old)

 

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BA/BSc (II Semester) Examination

MATHEMATICS

Paper : MATH–121 (Old Course)

( Calculus and Differential Equation )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer all questions.
2. The figures in the margin indicate fullmarks for the questions.

1. Answer any five questions of the following : 2×5=10

(a) Write the necessary and sufficient condition for the differential equation $Mdx+Ndy=0$ to be exact.

(b) Find the Wronskian of $x{e}^x$ and $e^x$ . Hence, conclude whether or not these are linearly independent.

(c) Explain nodes and cusps with suitable diagrams.

(d) Find the differential equation of the family of curves
$y=e^x(Acosx+Bsinx)$
where $A$ and $B$ are arbitrary constants.

(e) Find the order and the degree of the following differential equations :
(i) $y=\sqrt{x}\frac{dy}{dx}+\frac{k}{\left(\frac{dy}{dx}\right)}$
(ii) $k\left(\frac{d^{2}y}{d{x}^2}\right)={\left\{1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right\}}^{3/2.}$

(f) Find the complementary function for the differential equation
$D^3-4D^2+5D-2=x$ +1.

(g) Find the length of the circumference of the curve $x^2+y^2=1$.

2. Answer any four questions of the following : 5×4=20

(a) State and prove Leibnitz’s theorem for successive differentiation.

(b) Evaluate :
$\underset{x\to 0}{\lim }\frac{(1+x{)}^{1/x}-e}{x}$

(c) If the normal to the curve $x^{2/3}+y^{2/3}=a^{2/\mathrm{3 }}$makes an angle $\varphi$ with the axis of $x$, then show that its equation is $ycos\varphi -xsin\varphi =acos2\mathrm{\varphi .}$

(d) Prove that the radius of curvature for the Cartesian curve $y=f\left(x\right)$ at any point $(x,y)$ is given by
$\rho =\frac{{\left\{1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right\}}^{3/2}}{\left(\frac{d^{2}y}{d{x}^{2.}}\right)}$

(e) If $y_1\left(x\right)$ and $y_2\left(x\right)$ are any two solutions of $a_0\left(x\right)y\text{'}\text{'}\left(x\right)+a_1\left(x\right)y\text{'}\left(x\right)+a_2\left(x\right)y\left(x\right)=\mathrm{0 }$then prove that the linear combination $c_1{y}_1\left(x\right)+c_2{y}_2\left(\mathrm{x }\right)$where $c_1$ and $c_2$ are constants, is
also a solution of the given equation.

(f) Show that the general solution of the Clairaut’s equation $y=px+f\left(p\right)$ is $y=cx+f\left(c\right)$, where $c$ is an arbitrary constant.

(g) Solve the Lagrange’s equation $y=2px+p^2$.

(h) Find the perimeter of the cardioid $r=a(1-cos\theta )$.

3. Answer any five questions of the following : 10×5=50

(a) Prove that the two solutions $y_1(x)$and $y_2\left(x\right)$ of the equation
$a_0\left(x\right)y\text{'}\text{'}\left(x\right)+a_1\left(x\right)y\text{'}\left(x\right)+a_2\left(x\right)y\left(x\right)=0, a_0\left(x\right)\ne 0, x\in (a,b)$
are linearly dependent if and only if their Wronskian is identically zero.

(b) Let $I_{p,q}=\underset{0}{\overset{\pi /2}{\int }}{\sin }^{p}xco{s}^{q}xdx$, then prove that
$I_{p,q}=\frac{p-1}{p+q}{I}_{p-2,q}=\frac{q-1}{p+q}{I}_{p,q-2}$
where $p$ and $q$ are positive integers. Hence, evaluate
$\underset{0}{\overset{\pi /2}{\int }}{\sin }^{6}xco{s}^{8}xd\mathrm{x.}$

(c) (i) Evaluate
$\underset{0}{\overset{\pi /6}{\int }}{\cos }^{6}3\theta si{n}^{2}6\theta d\theta$
using reduction formula.
(ii) State the necessary and sufficient conditions for a function to be maximum or
minimum at a point. Show that $\sin x(1+cosx)$ is a maximum at $x=\frac{p}{3}$. 5+5

(d) (i) Show that the semi-vertical angle of the cone of maximum volume and of given slant height is ${\tan }^{-1}\left(\sqrt{2}\right)$.
(ii) If $y=x^2{e}^x$, then prove that
$\frac{d^{n}y}{d{x}^n}=\frac{1}{2}n(n-1)\frac{d^{2}y}{d{x}^2}-n(n-2)\frac{dy}{dx}+\frac{1}{2}(n-1)(n-2)y$. 5+5

(e) (i) Apply the method of variation of parameters to solve $y_2+4y=4tan\mathrm{x.}$
(ii) Reduce the differential equation $y\text{'}\text{'}+Py\text{'}+Qy=R$, where $P$, $\mathrm{Q }$and $R$ are functions of $x$, to the normal form $\frac{d^{2}v}{d{x}^2}+Iv=S$. 5+5

(f) (i) Find the asymptotes of the curve
$y^3-x^{2}y-2x{y}^2+2x^3-7xy+3y^2+2x^2+2x+2y+1=0$

(ii) Find the volume of the solid obtained by revolving the cardioid $r=a(1+cos\theta )$ about the initial line. 5+5

(g) Solve the following differential equations : 5+5
(i) $(1+y^2)dx=\left({\tan }^{-1}\left(y\right)-x\right)d\mathrm{y.}$
(ii) $\left(D^2+3D+2\right)y=e^x\sin x, where D\equiv \frac{d}{d\mathrm{x.}}$

(h) (i) Solve $xy\text{'}\text{'}-y\text{'}+4x^{3}y=x^5$.
(ii) Test for exactness and solve
$\frac{d^{3}y}{d{x}^3}+cosx\frac{d^{2}y}{{\mathrm{dx}}^2}-2sinx\frac{dy}{dx}-ycosx=sin2x$. 5+5

(i) Evaluate
$\underset{0}{\overset{1}{\int }}{x}^2{\left(1-x^2\right)}^{3/2}dx$
using reduction formula.