Q.1 a i) Solve $latex {(D^2-D)y = e^x sin (x)}&s=2$ Solution : The auxiliary equation is $latex {m^2-m =0}&s=1$ or $latex {m(m-1)=0}&s=1$. The roots are $latex {m_1 = 0}&s=1$ and $latex {m_2 = 1}&s=1$. The complimentary function will be $latex {y_c = c_1 e^{0x} + c_2 e^{1x} = c_1 + c_2 e^x}&s=2$ The particular integral… Continue reading M III CS/IT May 2016

# Category: M III

## Vector Integral Calculus

Vector Integration In the previous section, we discussed vector differentiation. We can extend our notion of integration of scalar functions to that of vector functions. Let $latex {\vec F (x,y,z)}&s=1$ be a vector field defined over a region and let $latex {C}&s=1$ be a curve in this region. At each point $latex {\vec F}&s=1$ will… Continue reading Vector Integral Calculus

## Statistics, Correlation and Regression

What is Statistics? It is a branch of mathematics. It involves collection, analysis, interpretation, presentation, and organization of data. (Dictionary Definition). Descriptive Statistics summarizes the data with the help of few indices, such as mean (the central tendency) and standard deviation (the dispersion). Inferential Statistics draws conclusions from data that are subject to random variation.… Continue reading Statistics, Correlation and Regression

## Partial Differential Equations

Introduction So far, while studying calculus, we have dealt with functions of single variable, i.e. $latex {y=f(x)}&s=1$. $latex {sin (x^2), ln \ x, e^{cos \ (tan \ x)}}&s=1$ are few examples. Irrespective of their complexity, the variable $latex {y}&s=1$ always depended on the value of independent variable $latex {x}&s=1$. We also defined the derivatives and… Continue reading Partial Differential Equations

## Applications of Linear Differential Equations to Electric Circuits

Prerequisites : Differential Equations of First Order and First Degree Linear Differential Equations of Higher Order There are 3 basic components of an electric circuit, where a change in voltage is possible. They are: 1) Resistance ($latex {R}&s=1$), voltage drop = $latex {i R}&s=1$, we saw this in Ohm's law. 2) Capacitance ($latex {C}&s=1$),… Continue reading Applications of Linear Differential Equations to Electric Circuits

## Simultaneous Linear Differential Equations

Introduction In the previous blospost, we covered the L.D.E.s of one dependent variable $latex {y}&s=1$ and one independent variable $latex {x}&s=1$. Moving ahead, we will now study L.D.E.s of two or more dependent variables and one independent variable. In order to solve such equations, we will need as many differential equations as the number of… Continue reading Simultaneous Linear Differential Equations

## Linear Differential Equations of Higher Order

Introduction An equation consisting of one (or more) independent variables and one (or more) dependent variables and at least one derivative is known as a differential equation (D.E.). The derivatives can be ordinary or partial. Depending on them, a D.E. can be ordinary D.E. or partial D.E. [Note the development so far: Set Theory, Relations,… Continue reading Linear Differential Equations of Higher Order