Diff. eq. peace group
- 1. Differential equation Sub Topic– Order,Degree and Formation of differential equation Presented By P.K Rai PGT (Maths), Peace Group K.V.Sector 8 R. K.Puram New Delhi
- 2. Learning objectives--- Students will be able to understand— • Definition of differential equation • Order and degree of a differential equation • General and particular solution • Formation of differential equation
- 3. Introduction -- Consider the function • -----(1) • Differentiating with respect to x we get dy/dx = 2x +5 ------(2) Consider another equations • -----(3) • ---(4) • ------(5) 2 5 6y x x= + + 2 3 3 0x x- + = sin cos 0x x+ = 7x y+ =
- 4. Definition • An equation involving derivative( derivatives) of dependent variable with respect to independent variable ( Variables) is called differential equations. • A differential equation involving derivative of dependent variable with respect to only one independent variable is called ordinary differential equation • Example.2 dy/dx+ x+y=0
- 5. Partial Differential equation - • An equation involving derivatives with respect to more than one independent variable s are called partial differential equations. • Note- we will use differential equation for ordinary differential in this chapter
- 6. Order of differential equation • Order of a differential is defined as the order of highest derivatives of the dependent variable with respect to independent variable.
- 7. Degree of differential equation - • Degree of a differential, when it is polynomial equation in derivatives ,is defined as the highest power of highest order derivatives .
- 8. Example 3 2 3 2 2 0 d y d y dy y dxdx dx æ ö ÷ç ÷+ - + =ç ÷ç ÷çè ø 2 2 sin 0 dy dy y dx dx æ ö ÷ç + - =÷ç ÷ç ÷è ø sin 0 dy dy dx dx æ ö ÷ç+ =÷ç ÷ç ÷è ø
- 9. General and particular solution of differential equation---- • Consider the differential equation ----(1) Consider the function y = a sin(x +b) --------(2) The equation (2) satisfies the differential equation(1).Hence function (2) is general solution of differential equation(1) 2 2 0 d y y dx + =
- 10. Particular solution-- • In the general solution y = a sin(x+b) ,if a and b are given some particular say a =2 and b= Then we get a function y = 2 sin(x+ ).This solution is called particular solution 4 p 4 p
- 11. Procedure to form differential equation that will represent a given family of curve-- • If the given family of curves depend on only one parameter then it is represented by --(1) Differentiating (1) with respect to x we get G(x,y,dy/dx,a) = 0 --(2) By eliminating a from equation from (1) and (2) we get an differential of the form F(x,y,dy/dx) = 0 1 ( , , ) 0F x y a =
- 12. If given family of curves F depends on the two parameter a,b(say) then it is represented by F(x,y,a,b) = 0 -(3) Differentiating it with respect x we get an equation of form g(x,y, dy/dx,a,b) =0 --(4) But it is not possible to eliminate a and bfrom two equation,so we need third equation. Differentiating (4) w.r.to x we get of the form h(x,y,dy/dx.d2y/dx2,a,b) = 0 (5) If we eliminate a and b from equation from 3,4 and 5 we get an equatio of the form (x, y, dy H dx
- 13. Example • 1. Form the differential equation representing the family of curves y = mx ,where m is constant • Sol. We have y = mx- (1) • Differentiating (1) with respect tox we get dy/dx = m -------(2) Putting the value of m in equation (1) we get y = dy/dx *x or xdy/dx –y = 0
- 14. Example • 2.Form a differential equation representing the family of curve x/a + y/b = 1 • Sol. Family of straight lines is x/a + y/b =1 -------(1) Differentiating eq. (1) with respect to x we get 1/a +1/b dy/dx =0 ----(2) Differentiating (2) with respect to x we get 0+1/b =0 or = 0 (ANS) 2 2 d y dx 2 2 d y dx
- 15. Evaluation tools--- Level1 –1. Find the order and degree of following differential equations (i) (ii) 2.Form the differential equation representing the family of curves 3 2 3 2 2 0 d y d y dy dxdx dx + + = 4 3 4 3 sin( ) 0 d y d y dx dx + = 2 2 2 ( )y a b x= -
- 16. Evaluation tools • Level 2.1. Form the differential equation of family of curves having centre at y-axis 2.Form the differential equation representing family of ellipse having foci on x-axis and cetre at origin 3.Verify that is a solution of 2x x y ae be x- = + + 2 2 2 2 2 d y dy x xy x o dxdx + - + - =
- 17. Hots 1.Prove that is the general solution of 2.Form the differential equation representing the family of curves given by 2 2 2 2 2 ( )x y c x y- = + 3 2 2 2 ( 3 ) ( 3 )x xy dx y x y dy- = - 2 2 2 ( ) 2x a y a- + =
- 18. • Thank • you