Ordinary differential equation

28 Feb 2019
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Ordinary differential equation

• 1. Introduction to Ordinary Differential Equations Presented by Dnyaneshwar Pardeshi [182070006] M.Tech (Control System) Under The Guidance Of Dr. Surendra Bhosale
• 2. Introduction  Differential equations are introduce in different fields and its importance appears not only in mathematics but also in Engineering , Natural science ,Chemical science , Medicine, Ecology and Economy.  In mathematics, an ordinary differential equation is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
• 3. Notation and Definitions • Differential equation: Equations, which are composed of an unknown function and its derivatives. 1. Ordinary Differential equation: When the function involves one independent variable, the equation is called an ordinary differential equation (or ODE). 2. Partial Differential equation: Differential equation involving two or more independent variables and its differentials.
• 4. Order : The order of highest derivative used in a differential equation Degree : Power of a highest derivative which is free from rationals & radical in the differential equation. 02 2  dt dy dt yd. 2nd order, 1st degree 3 3 dt xd x dt dx  3rd order, 1st degree
• 5. Solutions of First Order Differential Equations 1. Variable Separable Equations 2. Homogeneous & Non homogeneous Equation 3. Exact & Non-Exact Equations 4. Linear Equations 5. Bernoulli’s Equations
• 6. 1- Variable Separable Equations General form of differential equation is By seperating variable we get, By integrating we find the solution of this equation. 0),(),(  dyyxhdxyxf 0)()()()( 2121  dyyhxhdxyfxf 0 )( )( )( )( 2 1 2 1  dy yf yh dx xh xf
• 7. 2- Homogeneous Equation The condition of homogeneous function is where ‘n’ is degree of homogeneous function. Then Is the homogeneous differential equation if ‘f’ & ‘g’ are homogeneous function of same degree. Can be solved by reducing it into separable form by substitution i.e. ),(),( yxfyxf nn   ),( ),( yxg yxf dx dy  vxy  dx dv xv dx dy 
• 8. 3- Exact Equations The required condition for equation to be exact is and its general solution is [ is constant] [Terms free from ] 0),(),(  dyyxNdxyxM x N y M      cdyyxNdxyxM   ),(),( y x
• 9. 4. Linear Equations The standard form of linear DE is ...linear in y The integral factor that convert Linear Equations to exact equation is The general solution is )()( xQyxP dx dy   Pdx eFI. cdxFIQFIy   ).().(
• 10. 5. Bernoulli’s Equations I. Divide Bernoulli Equation over II. Put then This is linear equation & its solution as we told before n yxQyxP dx dy  )()( n y n yz   1 )()( 1 1 xQyxP dx dy y n n   dx dy ydx dz n n 1 )1( 1       )(1)(1 xQnzxPn dx dz 

Notes de l'éditeur

1. Partial differential equation that involves two or more independent variables.
2. These definations are same for both ode n pde