Math by Tathagata
Engineering Math Problems: In this page, you can find a list of Engineering Math Problems which can be useful for practice for final exam. Eng. Math II Content: Link Sequence, Series & Complex Analysis Q1: What is a bounded sequence. Show the sequence {xn} where $x_n=\dfrac{2n-1}{4n}$ is bounded. Q2: State ratio test and root test … Read more
In this page, lets learn Cauchy integral theorem and formula with examples. Cauchy’s Integral Theorem Statement If f(z) is continuous and analytic at each point within and on a closed curve C, then $\boxed{\displaystyle \int_C f(z)~dz=0}$. Cauchy’s Integral Formula If f(z) is analytic within and on a closed contour C and if $a$ is any … Read more
Cauchy-Euler equation (or Cauchy’s linear equation) is a differential equation with non-constant coefficients i.e, having variable-coefficients. It has the following form where a0, a1, …, an are real constants, and f(x) is a function of x. How to Solve Cauchy-Euler Equation? The below steps to be followed in order to solve a Cauchy-Euler equation. Solved … Read more
In this page, lets explore the variation of parameters method for solving a second order differential equation with examples. Variation of Parameters Formula Formula: Let u and v be two linearly independent solutions of y”+py’+qy=0. Then the general solution of y”+py’+qy=R obtained by using the method of variations of parameters is given by $y=u \displaystyle\int … Read more
Form ODE by Eliminating a and b: In this article, let’s learn how to form an ordinary differential equation (ODE) by eliminating a and b which are arbitrary constants. Lets solve some problems. Solved Problems $\boxed{\color{blue}\textbf{Q}1:}$ Eliminate the arbitrary constants a, b and c to from an ordinary differential equation. $\boxed{\color{red}\textbf{Answer:}}$ Part (1) Given y … Read more
On this page, we solve few problems on the topic of the change of variables in double integrals using Jacobian method. Questions- Answers on Change of Variables $\boxed{\color{blue}\textbf{Q 1}:}$ Evaluate the double integral I = $\displaystyle \int \displaystyle \int_R x dx dy$ where R = $\{(x,y) \in \mathbb{R}^2: 1 \leq x(1-y)\leq 2, 1 \leq xy … Read more
On this page, we solve some integrals using the change of order of integration. Solved Problems $\boxed{\color{blue}\textbf{Q 1}:}$ Evaluate $\displaystyle \int_{x=0}^1 \displaystyle \int_{y=0}^{\sqrt{1-x^2}} \sqrt{1-y^2} dy dx$ Answer: We will use the change of order of integration. Note R:= $\begin{cases} 0 \leq y \leq \sqrt{1-x^2} \\ 0 \leq x \leq 1. \end{cases}$ So 0 ≤ y2 … Read more
On this page, you will find Engineering Mathematics practice problems. Study Materials: nth Derivative Coplanar Vectors Indeterminate Forms and L’Hospital Rule Characteristic Equation of a Matrix Rolle’s Theorem Cayley-Hamilton Theorem Taylor Series Expansion Volume generated by revolving a curve Eulers Theorem for Homogeneous Functions Area generated by revolving a curve Vector Algebra Change of Order … Read more
Coplanar vectors lie on the same plane. In this page , we learn about coplanar vectors, conditions to be coplanar along with some solved and practice problems. Coplanar Vectors Definition: A set of vectors are said to be coplanar if they lie on the same plane. In other words, any linear combination of this set … Read more
Euler’s theorem for homogeneous functions states that for a homogeneous function of degree n, the sum of each variable multiplied by its partial derivative equals n times the function itself. Homogeneous Function Definition: A function f(x,y) is said to be homogeneous of degree n if f(tx,ty) = tn f(x,y) for all real numbers t. Euler’s … Read more