Topics
Beta and Gamma Functions, Differentiation Under Integral Sign and Exact Differential Equation old
Beta and Gamma Functions and Its Properties
Rectification of Plane Curves
Differential Equation of First Order and First Degree
Differential Calculus old
Linear Differential Eqaution with Constant Coeffiecient
Linear Differential Equations(Review), Equation Reduciable to Linear Form, Bernoulli’S Equation
Cauchy’S Homogeneous Linear Differential Equation and Legendre’S Differential Equation, Method of Variation of Parameters
Simple Application of Differential Equation of First Order and Second Order to Electrical and Mechanical Engineering Problem
Numerical Solution of Ordinary Differential Equations of First Order and First Degree and Multiple Integrals old
Multiple Integrals‐Double Integration
Taylor’S Series Method,Euler’S Method,Modified Euler Method,Runga‐Kutta Fourth Order Formula
Multiple Integrals with Application and Numerical Integration old
Triple Integration
Application to Double Integrals to Compute Area, Mass, Volume. Application of Triple Integral to Compute Volume
Numerical Integration
Differential Equations of First Order and First Degree
- Exact Differential Equations
- Equations Reducible to Exact Form by Using Integrating Factors
- Linear Differential Equations
- Equations Reducible to Linear Equations
- Bernoulli’S Equation
- Simple Application of Differential Equation of First Order and First Degree to Electrical and Mechanical Engineering Problem
Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
- Linear Differential Equation with Constant Coefficient‐ Complementary Function
- Particular Integrals of Differential Equation
- Cauchy’S Homogeneous Linear Differential Equation
- Legendre’S Differential Equation
- Method of Variation of Parameters
Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
- Taylor’S Series Method
- Euler’S Method
- Modified Euler Method
- Runga‐Kutta Fourth Order Formula
- Beta and Gamma Functions and Its Properties
Differentiation Under Integral Sign, Numerical Integration and Rectification
- Differentiation Under Integral Sign with Constant Limits of Integration
- Numerical Integration‐ by Trapezoidal
- Numerical Integration‐ by Simpson’S 1/3rd
- Numerical Integration‐ by Simpson’S 3/8th Rule
- Rectification of Plane Curves
Double Integration
- Double Integration‐Definition
- Evaluation of Double Integrals
- Change the Order of Integration
- Evaluation of Double Integrals by Changing the Order of Integration and Changing to Polar Form
Triple Integration and Applications of Multiple Integrals
- Triple Integration Definition and Evaluation
- Application of Double Integrals to Compute Area
- Application of Double Integrals to Compute Mass
- Application of Double Integrals to Compute Volume
- Application of Triple Integral to Compute Volume
Notes
A differential equation of the from `(dy)/(dx) + Py = Q`
To solve the first order linear differential equation of the type
`(dy)/(dx) + Py= Q` ...(1)
Multiply both sides of the equation by a function of x say g (x) to get
g(x)`(dy)/(dx) + P.(g(x)) y = Q . g(x)` ...(2)
Choose g(x) in such a way that R.H.S. becomes a derivative of y . g (x).
i.e. `g(x)(dy)/(dx) + P.g(x)y = d/(dx) [y.g(x)]`
or `g(x) (dy)/(dx) + P.g(x)y ` = g(x)`(dy)/(dx) + y g'(x)`
`=> P.g(x) = g'(x)`
o r P = g'(x)/g(x)
Integrating both sides with respect to x, we get
`int Pdx = int (g'(x))/g(x)dx`
or `int P.dx = log (g(x))`
or g(x) = `e^(int Pdx)`
On multiplying the equation (1) by g(x) =`e^( int Pdx)` , the L.H.S. becomes the derivative of some function of x and y. This function
g(x) = `e^(int P dx)` is called Interrating Factor (I.F.) of the given differential equation.
Substituting the value of g (x) in equation (2), we get
`e^(Pdx) (dy)/(dx) + Pe^(int Pdx) y = Q . e^(Pdx)`
Or `d/(dx) (ye^(intPdx)) = Qe^(int Pdx)`
Integrating both sides with respect to x, we get
`y.e^(int P dx) = int (Q.e^(int P dx)) dx` Or
`y = e^(-int Pdx) = int (Q.e^(int P dx)) dx + C`
which is the general solution of the differential equation.
Steps involved to solve first order linear differential equation:
(i) Write the given differential equation in the form `(dy)/(dx)` + Py = Q where P, Q are constants or functions of x only.
(ii) Find the Integrating Factor (I.F) = `e^(int Pdx)`
(iii) Write the solution of the given differential equation as
y (I.F) = `int`(Q × I.F )dx + C
In case, the first order linear differential equation is in the form `(dx)/(dy) + P_1x = Q_1`, where, `P_1` and `Q_1` are constants or functions of y only.
Then I.F = `e^(P_idy)` and the solution of the differential equation is given by
x . (I.F) = `int (Q_1 xx I.F)` dy + C
