B.Sc. Mathematics Honours C-4 Question Paper 2018 (CBCS)

Unit-I

[Marks: 22]

1. Answer any one question :
(a) Let $y = \phi(x)$ be a solution for $0 < x < \alpha$ of the Euler equation $x^{2}y'' + axy' + by = 0$ where $a, b$ are constants. Let $\psi(t) = \phi(e^{t})$, then show that $y$ satisfies the equation $$\frac{d^{2}\psi}{dt^{2}} + (a-1)\frac{d\psi}{dt} + b\psi = 0$$
(b) Test whether the solutions $e^{x}, e^{2x}, e^{3x}$ are linearly independent or not.

2. Answer any two questions :
(a) Knowing that $y = x$ is a solution of the equation $x^{2}\frac{d^{2}y}{dx^{2}} - x(x+2)\frac{dy}{dx} + (x+2)y = 0 (x \ne 0)$, reduce the equation $x^{2}\frac{d^{2}y}{dx^{2}} - x(x+2)\frac{dy}{dx} + (x+2)y = x^{3} (x \ne 0)$ to a differential equation of first order and first degree and find its complete primitive.
(b) Solve the differential equation : $$\frac{d^{3}y}{dx^{3}} + 2\frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} = e^{-x} + \cos x$$
(c) Solve the equation $$\frac{d^{2}y}{dx^{2}} + a^{2}y = \tan ax$$ by the method of variation of parameters.

3. Answer any one question :
(a) (i) Solve the differential equation $$\frac{d^{2}y}{dx^{2}} + 4y = x^{2} \sin 2x$$ by the method of undetermined co-efficients.
(ii) State the sufficient condition for existence and uniqueness of the solution of the differential equation $\frac{dy}{dx} = f(x, y)$, $y(x_{0}) = y_{0}$. Show that $\frac{dy}{dx} = \frac{1}{y}$, $y(0) = 0$ has more than one solution and indicate the possible reason.
(b) (i) Let $a_{1}, a_{2}$ are continuous functions on $[a, b]$ and $\phi_{1}, \phi_{2}$ be the two independent solutions of $y''(x) + a_{1}(x)y'(x) + a_{2}(x)y(x) = 0$ on some interval $[a, b]$. Let $x_{0}$ be any point in $[a, b]$. Then show that $$W(\phi_{1}, \phi_{2})(x) = \exp \left\{ -\int_{x_{0}}^{x} a_{1}(t) dt \right\} W(\phi_{1}, \phi_{2})(x_{0})$$ for all $x \in [a, b]$ where $W(\phi_{1}, \phi_{2})(x) = \det \begin{pmatrix} \phi_{1}(x) & \phi_{2}(x) \\ \phi_{1}'(x) & \phi_{2}'(x) \end{pmatrix}$.
(ii) Solve the differential equation $$x^{2}\frac{d^{2}y}{dx^{2}} + 3x\frac{dy}{dx} + y = \frac{1}{(1-x)^{2}}$$

Unit-II

[Marks: 13]

4. Answer any four questions :
(a) Solve the equations $\frac{dx}{dt} = -\omega y$ and $\frac{dy}{dt} = \omega x$ and show that the point $(x, y)$ lies on a circle.
(b) Solve the equation $$\frac{dx}{x^{2}-y^{2}-z^{2}} = \frac{dy}{2xy} = \frac{dz}{2xz}$$
(c) Find the complementary function for the system $(D+3)x + Dy = \cos t$ and $(D-1)x + y = \sin t$ where $D \equiv \frac{d}{dt}$.
(d) Solve : $\frac{yz dx}{y-z} = \frac{zx dy}{z-x} = \frac{xy dz}{x-y}$
(e) Show that the solution of the differential equations $\frac{dx}{dt} = 2x + y$ and $\frac{dy}{dt} = 3x$ satisfies the relation $3x + y = ke^{3t}$ where $k$ is a real constant.
(f) If $\frac{dy_{1}}{dx} = 3y_{1} + 4y_{2}$ and $\frac{dy_{2}}{dx} = 4y_{1} + 3y_{2}$ then find the value of $y_{1}(x)$.

5. Answer any one question :
(a) Find the fundamental matrix and the complementary solution of the homogeneous linear system of differential equations $\frac{dx}{dt} = 3x + y$ and $\frac{dy}{dt} = x + 3y$.
(b) (i) Solve the equation $(x^{2} + y^{2} + z^{2})dx - 2xydy - 2xzdz = 0$.
(ii) Find $f(y)$ such that the total differential $\frac{yz+z}{x}dx - zdy + f(y)dz = 0$ is integrable. Hence solve it.

Unit-III

[Marks: 9]

6. Answer any two questions :
(a) Consider the set of non-linear differential equations $\frac{dx}{dt} = x - xy$; $\frac{dy}{dt} = -y + xy$. Find the equilibrium points of the system of equations.
(b) Show that $x = 0$ is an ordinary point and $x = 1$ is a regular singular point of the ODE $$x(x-1)\frac{d^{2}y}{dx^{2}} + \sin x \frac{dy}{dx} + 2x(x-1)y = 0$$
(c) What do you mean by stable and unstable critical points.

7. Answer any one question :
(a) Find the phase curve of the system of dynamical equations $\dot{x} = -x - 2y$ and $\dot{y} = 2x - y$. Also show that the system is stable.
(b) Find the power series solution of the equation $(x^{2} + 1)\frac{d^{2}y}{dx^{2}} + x\frac{dy}{dx} - xy = 0$ in power of $x$ about the origin.

Unit-IV

[Marks: 16]

8. Answer any three questions :
(a) Show that the vector $\vec{F} = (2x-yz)\hat{i} + (2y-zx)\hat{j} + (2z-xy)\hat{k}$ is irrotational.
(b) Test the continuity of the vector function $\vec{f}(t) = |t|\hat{i} - \sin t \hat{j} + (1 + \cos t)\hat{k}$ at $t = 0$.
(c) If $\vec{A} = (3x^{2} + 6y)\hat{i} - 14yz\hat{j} + 20z^{2}x\hat{k}$, evaluate $\int_{c} \vec{A} \cdot d\vec{r}$ from $(0, 0, 0)$ to $(1, 1, 1)$ along the path $c : x = t, y = t^{2}, z = t^{3}$.
(d) Find the unit vector in the direction of the tangent at any point on the curve given by $\vec{r} = (a \cos t)\hat{i} + (a \sin t)\hat{j} + bt\hat{k}$.
(e) Show that the vectors $\vec{a} \times (\vec{b} \times \vec{c}), \vec{b} \times (\vec{c} \times \vec{a}), \vec{c} \times (\vec{a} \times \vec{b})$ are coplanar.

9. Answer any one question :
(a) (i) If $\vec{r} = \vec{a} \cos nt + \vec{b} \sin nt$, where $a, b, n$ are constants, then prove that $\frac{d^{2}\vec{r}}{dt^{2}} + n^{2}\vec{r} = \vec{0}$ and $\vec{r} \times \frac{d\vec{r}}{dt} = n(\vec{a} \times \vec{b})$.
(ii) Derive the volume of a tetrahedron whose co-ordinates of vertices are given. Use it to calculate the volume of the tetrahedron whose vertices are $A(2,-1,-3), B(4,1,3), C(3,3,-1)$ and $D(1,4,2)$.
(b) (i) Prove that $[(\vec{\alpha} \times \vec{\beta}), (\vec{\beta} \times \vec{\gamma}), (\vec{\gamma} \times \vec{\alpha})] = [\vec{\alpha}, \vec{\beta}, \vec{\gamma}]^{2}$ where $[\cdot]$ denotes the scalar triple product.
(ii) Find $\vec{t}, \vec{n}, \vec{b}$ for the curve given by $\vec{r} = (e^{t} \cos t, e^{t} \sin t, e^{t})$ at $t = 0$.