Engineering Mathematics-II

Cheatsheet Content

### Ordinary Differential Equations (ODEs) - **Linear ODEs of nth Order with Constant Coefficients:** - Form: $a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} + ... + a_1 \frac{dy}{dx} + a_0 y = Q(x)$ - Auxiliary Equation: $a_n m^n + a_{n-1} m^{n-1} + ... + a_1 m + a_0 = 0$ - Complementary Function (CF) depends on roots of auxiliary equation: - Real and distinct roots $m_1, m_2, ...$: $y_c = C_1 e^{m_1 x} + C_2 e^{m_2 x} + ...$ - Real and repeated roots $m, m, ...$ (k times): $y_c = (C_1 + C_2 x + ... + C_k x^{k-1}) e^{mx}$ - Complex conjugate roots $\alpha \pm i\beta$: $y_c = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x))$ - Particular Integral (PI) methods: Undetermined Coefficients, Variation of Parameters. - **Simultaneous Linear Differential Equations:** - System of equations involving derivatives of multiple dependent variables with respect to a single independent variable. - Can be solved using elimination method or operator method. - **Second Order Linear ODEs with Variable Coefficients:** - Form: $P(x) \frac{d^2 y}{dx^2} + Q(x) \frac{dy}{dx} + R(x) y = S(x)$ - **Solution by Changing Independent Variable:** If one solution $y_1(x)$ is known, assume $y = v(x)y_1(x)$ and substitute. - **Method of Variation of Parameters:** For $y'' + P(x)y' + Q(x)y = R(x)$, if $y_1, y_2$ are solutions to the homogeneous equation, then $y_p = u_1 y_1 + u_2 y_2$ where $u_1' = -\frac{y_2 R}{W}$ and $u_2' = \frac{y_1 R}{W}$, and $W = y_1 y_2' - y_2 y_1'$ (Wronskian). - **Cauchy-Euler Equation (Homogeneous Linear Equation with Variable Coefficients):** - Form: $a_n x^n \frac{d^n y}{dx^n} + a_{n-1} x^{n-1} \frac{d^{n-1} y}{dx^{n-1}} + ... + a_1 x \frac{dy}{dx} + a_0 y = Q(x)$ - Substitute $x = e^t$ or $t = \ln x$, then $x \frac{dy}{dx} = D_t y$, $x^2 \frac{d^2 y}{dx^2} = D_t(D_t-1)y$, etc., where $D_t = \frac{d}{dt}$. This transforms it into an ODE with constant coefficients. - **Applications:** Modeling physical systems (e.g., spring-mass systems, RLC circuits). ### Laplace Transform - **Definition:** $\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t) dt$ - **Existence Theorem:** Conditions for existence (e.g., $f(t)$ is piecewise continuous and of exponential order). - **Properties:** - Linearity: $\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$ - First Shifting Theorem: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ - Second Shifting Theorem: $\mathcal{L}\{f(t-a)U(t-a)\} = e^{-as}F(s)$ - Scaling Property: $\mathcal{L}\{f(at)\} = \frac{1}{a}F(\frac{s}{a})$ - Differentiation of Transform: $\mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n}{ds^n} F(s)$ - Integration of Transform: $\mathcal{L}\{\frac{f(t)}{t}\} = \int_s^\infty F(u) du$ - **Laplace Transform of Derivatives:** - $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$ - $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$ - $\mathcal{L}\{f^{(n)}(t)\} = s^n F(s) - s^{n-1}f(0) - ... - f^{(n-1)}(0)$ - **Laplace Transform of Integrals:** - $\mathcal{L}\{\int_0^t f(\tau) d\tau\} = \frac{F(s)}{s}$ - **Unit Step Function (Heaviside Function):** $U(t-a) = \begin{cases} 0 & t ### Sequence and Series - **Definition of Sequence:** An ordered list of numbers $a_1, a_2, a_3, ...$. - **Definition of Series:** The sum of the terms of a sequence, $\sum_{n=1}^\infty a_n$. - **Convergence of Series:** A series converges if its sequence of partial sums converges to a finite limit. - **Tests for Convergence of Series:** - **Ratio Test:** For $\sum a_n$, let $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. - If $L 1$ or $L = \infty$, series diverges. - If $L = 1$, test is inconclusive. - **D'Alembert's Test (same as Ratio Test):** - **Raabe's Test:** If Ratio Test is inconclusive ($L=1$), let $L = \lim_{n \to \infty} n(|\frac{a_n}{a_{n+1}}| - 1)$. - If $L > 1$, series converges. - If $L 0$ (finite), then $\sum a_n$ and $\sum b_n$ either both converge or both diverge. - **Integral Test:** If $f(x)$ is positive, continuous, and decreasing for $x \ge N$, then $\sum a_n$ and $\int_N^\infty f(x)dx$ either both converge or both diverge. - **P-series Test:** $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$. - **Alternating Series Test:** For $\sum (-1)^n b_n$, if $b_n > 0$, $b_n$ is decreasing, and $\lim_{n \to \infty} b_n = 0$, then the series converges. - **Fourier Series:** Represents a periodic function as a sum of sines and cosines. - For a function $f(x)$ with period $2L$ on $[-L, L]$: $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos(\frac{n\pi x}{L}) + b_n \sin(\frac{n\pi x}{L}))$ where $a_0 = \frac{1}{L} \int_{-L}^L f(x) dx$ $a_n = \frac{1}{L} \int_{-L}^L f(x) \cos(\frac{n\pi x}{L}) dx$ $b_n = \frac{1}{L} \int_{-L}^L f(x) \sin(\frac{n\pi x}{L}) dx$ - **Half-Range Fourier Sine Series:** For a function defined on $[0, L]$, extended as an odd function to $[-L, L]$. Only sine terms are present ($a_n = 0$). $f(x) = \sum_{n=1}^\infty b_n \sin(\frac{n\pi x}{L})$ where $b_n = \frac{2}{L} \int_0^L f(x) \sin(\frac{n\pi x}{L}) dx$. - **Half-Range Fourier Cosine Series:** For a function defined on $[0, L]$, extended as an even function to $[-L, L]$. Only cosine terms are present ($b_n = 0$). $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(\frac{n\pi x}{L})$ where $a_0 = \frac{2}{L} \int_0^L f(x) dx$ and $a_n = \frac{2}{L} \int_0^L f(x) \cos(\frac{n\pi x}{L}) dx$. ### Complex Variable - Differentiation - **Functions of Complex Variable:** $w = f(z) = u(x,y) + iv(x,y)$, where $z = x+iy$. - **Limit:** $\lim_{z \to z_0} f(z) = L$. - **Continuity:** $f(z)$ is continuous at $z_0$ if $\lim_{z \to z_0} f(z) = f(z_0)$. - **Differentiability:** $f(z)$ is differentiable at $z_0$ if $f'(z_0) = \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z}$ exists and is unique, regardless of the direction $\Delta z \to 0$. - **Analytic Functions:** A function $f(z)$ is analytic at a point $z_0$ if it is differentiable not only at $z_0$ but at every point in some neighborhood of $z_0$. - **Cauchy-Riemann Equations:** Necessary conditions for differentiability. - **Cartesian Form:** If $f(z) = u(x,y) + iv(x,y)$ is analytic, then $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$. - **Polar Form:** If $f(z) = u(r,\theta) + iv(r,\theta)$ is analytic, then $\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta}$ and $\frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta}$. - **Harmonic Function:** A function $\phi(x,y)$ is harmonic if it satisfies Laplace's equation: $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$. - If $f(z) = u+iv$ is analytic, then both $u$ and $v$ are harmonic functions. $u$ and $v$ are called harmonic conjugates. - **Method to find Analytic Functions:** - **Milne's Thompson Method:** If $u(x,y)$ is given, replace $x$ with $z$ and $y$ with $0$ in $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ to find $\phi_1(z,0)$ and $\phi_2(z,0)$. Then $f'(z) = \phi_1(z,0) - i\phi_2(z,0)$. Integrate to find $f(z)$. (Similar for $v(x,y)$). - **Conformal Mapping:** A transformation $w = f(z)$ is conformal if it preserves angles between curves, both in magnitude and sense. An analytic function $f(z)$ is conformal at points where $f'(z) \ne 0$. - **Mobius Transformation (Bilinear Transformation):** - Form: $w = \frac{az+b}{cz+d}$ where $ad-bc \ne 0$. - Maps circles and lines to circles and lines. - Preserves cross-ratio. ### Complex Variable - Integration - **Complex Integration (Contour Integral):** $\int_C f(z) dz$. - If $f(z) = u+iv$ and $dz = dx+idy$, then $\int_C f(z) dz = \int_C (u dx - v dy) + i \int_C (v dx + u dy)$. - **Cauchy-Integral Theorem:** If $f(z)$ is analytic within and on a simple closed contour $C$, then $\oint_C f(z) dz = 0$. - **Cauchy Integral Formula:** If $f(z)$ is analytic within and on a simple closed contour $C$ and $a$ is any point inside $C$, then $f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} dz$. - For derivatives: $f^{(n)}(a) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z-a)^{n+1}} dz$. - **Taylor's Series:** If $f(z)$ is analytic inside a circle $C$ with center $z_0$, then for every point $z$ inside $C$: $f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$. - **Laurent's Series:** If $f(z)$ is analytic in an annulus $R_1