Q: Q.9 Classify the following equation -

The given three-dimensional Laplace equation $\nabla^2 u = 0$ is rigorously classified entirely as an Elliptic Partial Differential Equation since it describes a purely spatial steady-state condition completely lacking any real characteristics.

Question 1

**PART - A**

Accepted Answer

The Rank-Nullity Theorem mathematically asserts that for any linear transformation strictly mapping a finite-dimensional vector space exactly to another, the sum of the transformation's rank (dimension of the image) and its nullity (dimension of the kernel) rigorously equals the total dimension of the domain.

Question 2

Q.2 Define orthogonal matrix.

Accepted Answer

An orthogonal matrix is rigorously defined as a real square matrix whose mathematical transpose is absolutely exactly equal to its inverse, strictly satisfying the fundamental condition $A A^T = A^T A = I$.

Question 3

Q.3 Write the Integrating Factor (I.F.) of the following differential equation -

Accepted Answer

By rigorously converting the given differential equation strictly into the standard linear form $\frac{dx}{dy} + Px = Q$, the exact mathematical Integrating Factor (I.F.) is explicitly evaluated as $e^{	an^{-1} y}$.

Question 4

Q.4 Write the Clairaut's form of ordinary differential equation.

Accepted Answer

Clairaut's equation is a highly specific first-order differential equation rigidly taking the exact mathematical form $y = px + f(p)$, where $p = \frac{dy}{dx}$, whose absolute general solution is instantly found by replacing $p$ with a constant $c$.

Question 5

Q.5 Solve : $(D^2 - 3D + 2) y = e^x$

Accepted Answer

By systematically evaluating the complementary function from the auxiliary roots $m=1, 2$ and calculating the particular integral explicitly for $D=1$ using the derivative rule, the complete exact solution is rigorously found to be $y = C_1 e^x + C_2 e^{2x} - x e^x$.

Question 6

Q.6 Define power series.

Accepted Answer

A mathematical power series is rigorously defined as an infinite polynomial series explicitly taking the standard form $\sum_{n=0}^{\infty} c_n(x-a)^n$, utilized fundamentally to represent highly complex functions entirely within its absolute radius of convergence.

Question 7

Q.7 Form the partial differential equation, given that $z = a(x + y) + b$.

Accepted Answer

By explicitly taking the first-order partial derivatives of the given function $z = a(x + y) + b$ with respect exactly to $x$ and $y$, the resulting partial differential equation is rigorously proven to be strictly $p = q$.

Question 8

Q.8 Solve : $a(p + q) = z$

Accepted Answer

By systematically solving the exact auxiliary equations for the given Lagrange's linear PDE $ap + aq = z$, the two independent integral surfaces are extracted, yielding the absolute general solution $f(x-y, z e^{-x/a}) = 0$.

Question 9

Q.9 Classify the following equation -

Accepted Answer

The given three-dimensional Laplace equation $
abla^2 u = 0$ is rigorously classified entirely as an Elliptic Partial Differential Equation since it describes a purely spatial steady-state condition completely lacking any real characteristics.

Question 10

Q.10 Write the one dimensional wave equation.

Accepted Answer

The standard one-dimensional wave equation is rigorously defined mathematically as $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$, representing physical wave propagation strictly moving exactly at speed $c$.

Question 11

**PART - B**

Accepted Answer

By systematically applying elementary row and column transformations, the given $3 	imes 4$ matrix is rigorously reduced precisely to its canonical normal form $[I_3 \quad 0]$, mathematically proving its exact rank is $3$.

Question 12

Q.2 Solve : $y = 2px + p^2 y$

Accepted Answer

By systematically substituting $p = \frac{dy}{dx}$ and rigorously differentiating the entire equation exactly with respect to $y$, the differential equation is mathematically solved to strictly yield the general solution $y^2 = 2cx + c^2$.

Question 13

Q.3 Solve : $(D^2 + 3D + 2) y = e^{2x} \sin x$

Accepted Answer

By rigorously calculating the Complementary Function (C.F.) and systematically applying the exponential shift theorem to deduce the Particular Integral (P.I.), the absolute general solution is mathematically computed as $y = C_1 e^{-x} + C_2 e^{-2x} + \frac{e^{2x}}{170}(11 \sin x - 7 \cos x)$.

Question 14

Q.4 Solve : $\sin^2 x \frac{d^2y}{dx^2} - 2y = 0$

Accepted Answer

By recognizing a known solution $y_1 = \cot x$ and systematically applying the method of reduction of order (or variation of parameters), the complete general solution is mathematically derived as $y = C_1 \cot x + C_2 (1 - x \cot x)$.

Question 15

Q.5 Solve : $\frac{dx}{z-y} = \frac{dy}{x-z} = \frac{dz}{y-x}$

Accepted Answer

By systematically employing two distinct sets of highly specific multipliers $(1, 1, 1)$ and $(x, y, z)$ directly on Lagrange's auxiliary equations, two absolutely independent integrals are formed, establishing the final solution exactly as $f(x+y+z, x^2+y^2+z^2) = 0$.

Question 16

Q.6 Solve : $9 (p^2 z + q^2) = 4$

Accepted Answer

By systematically introducing the specific transformation variable $u = x + ay$ to rigorously convert the non-linear PDE entirely into an ordinary differential equation, the final complete integral is mathematically computed exactly as $(z+a^2)^3 = (x + ay + c)^2$.

Question 17

Q.7 Using the method of separation of variables, solve -

Accepted Answer

By systematically applying the Method of Separation of Variables assuming $u(x,t) = X(x)T(t)$ and matching against the exact boundary condition $u(x,0) = 6e^{-3x}$, the exact physical solution is mathematically proven strictly to be $u(x,t) = 6e^{-3x - 2t}$.

Question 18

**PART - C**

Accepted Answer

By systematically calculating the characteristic polynomial $\lambda^3 - 6\lambda^2 + 7\lambda + 2 = 0$, the Cayley-Hamilton theorem $A^3 - 6A^2 + 7A + 2I = 0$ is rigorously verified using matrix multiplication, and $A^{-1}$ is exactly computed from this identity.

Question 19

Q.2 Solve -

Accepted Answer

By systematically determining the exact integrating factor $1/(xM - yN) = \frac{1}{2x^3 y^3}$ for the given homogeneous-type equation, the ODE is rendered exact and mathematically solved to yield $\frac{x^2y^2}{4} + \ln x - \ln y - \frac{1}{2xy} = C$.

Question 20

Q.3 Apply the method of variation of parameter to solve -

Accepted Answer

By rigorously deploying the Method of Variation of Parameters, the Wronskian is calculated as $-2$, enabling the exact integration of the parameter functions $u(x)$ and $v(x)$ to yield the complete mathematical solution $y = C_1 e^x + C_2 e^{-x} - 1 - e^x \ln(1+e^{-x}) + e^{-x} \ln(1+e^x)$.

Question 21

Q.4 Apply Charpit's method to solve -

Accepted Answer

By systematically deploying Charpit's method and rigidly extracting the algebraic relation $p = aq$ from the highly complex auxiliary equations, the non-linear PDE is completely solved, yielding the absolute complete integral $z = ax + \frac{a x y}{a-1} + c$.

Question 22

Q.5 Discuss the solution of two dimensional Laplace's equation.

Accepted Answer

The highly critical 2D Laplace equation $
abla^2 u = 0$ is rigorously solved entirely via the Method of Separation of Variables, systematically producing exact mathematical solutions involving specific combinations of trigonometric (sine/cosine) and hyperbolic/exponential functions rigidly dependent on the physical boundary condition parameter $k$.

Mathematics II