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Maths differential equation ppt
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- 3. DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATION Anequation involving one dependent variable, one or more independent variables and the differential coefficients (derivatives) of dependent variable with respect to independent variables is called a differential equation.
- 4. DIFFERENTIAL EQUATIONS Examples i) d𝟐 y dx𝟐 + y dy dx =3 ii) ∂u ∂x + ∂u ∂y = 2(x+y)
- 5. DIFFERENTIAL EQUATIONS Types ofdifferential equations There are two types of differential equations i) Ordinary differential equations ii) Partial differential equations
- 6. DIFFERENTIAL EQUATIONS ORDINARY DIFFERENTIALEQUATION A differential equation is said to be an ordinary differential equation if it contains only one independent variable and ordinary derivatives with respect to this independent variable. i) x dy dx + 3y = 0 ii) 4x d𝟐 y dx𝟐 + dy dx − 2y𝟐 = 0
- 7. DIFFERENTIAL EQUATIONS PARTIAL DIFFERENTIALEQUATION A differential equation is said to be a partial differential equation if it contains at least two independent variables and partial derivatives with respect to either of these independent variables. Examples i) x ∂u ∂x + y ∂u ∂y = u ii) x𝟐 ∂𝟐u ∂x 𝟐 + xy ∂𝟐u ∂x∂y + y𝟐 ∂𝟐u ∂y 𝟐 = 3u
- 8. DIFFERENTIAL EQUATIONS Order: The orderof the differential equation is the order of the highest order derivative involving in it. Examples i) The order of dy dx 𝟑/𝟐 = x𝟐 is 1 ii) The order of x2 d 𝟐 y dx 𝟐 + x+ dy dx 𝟑 +5 = 0 is 2
- 9. DIFFERENTIAL EQUATIONS Degree: The degreeof the highest derivative involved in an ordinary differential equation, when the equation has been expressed in the form of a polynomial in the highest derivative by eliminating radicals and fractional powers of the derivatives is called the degree of the differential equation.
- 10. DIFFERENTIAL EQUATIONS Example i) Theorder of d 𝟐 y dx 𝟐 + 2 x+ dy dx 𝟑 +5 = 0 is 1 ii) The order of 5+ dy dx 𝟐 𝟓/𝟑 = 2 d 𝟐 y dx 𝟐 is 3
- 11. DIFFERENTIAL EQUATIONS Note: The degreeof some differential equations may not be defined. Example i) The degree of edy/dx = dy dx +y is not defined since the differential equation can not be expressed as a polynomial in dy dx .
- 12. DIFFERENTIAL EQUATIONS 1) Theorder of the highest order derivative involved in a differential equation is called _______ 1) degree of D.E 2) order of D.E 3) both order and degree of D.E 4) neither order nor degree of D.E
- 13. DIFFERENTIAL EQUATIONS 2) Theorder of the differential equation d 𝟑 y dx𝟑 -5 dy dx +6y=0 is_______ 1) 1 2) 2 3) 3 4) 4
- 14. DIFFERENTIAL EQUATIONS 3) Thedegree of the differential equation d 𝟒 y dx𝟒+2 dy dx 𝟒 =cosx is_____ 1) 3 2) 2 3) 4 4) 1
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- 16. DIFFERENTIAL EQUATIONS Formation ofa differential equation An ordinary differential equation is obtained when we eliminate arbitrary constants (also called parameters) from a given relation involving the variables; The order of the differential equation being equal to the number of the arbitrary constants to be eliminated.
- 17. DIFFERENTIAL EQUATIONS Let(x,y,c1,c2,c3,…..cn) = 0 (1) be a relation involving variables x,y and n arbitrary constants c1,c2,….cn. If we differentiate n times w.r.t ‘x’ and eliminate the n arbitrary constants from the (n+1) relations, we get an ordinary differential equation of the form F(x,y,y1,y11,y111.…y(n) ) = 0 Where y1 = dy dx , y11 = d 𝟐 y dx 𝟐…..y(n) = d 𝒏 y dx 𝒏
- 18. DIFFERENTIAL EQUATIONS Example: Findthe differential equation of the family of curves y = Ae2x+Be-2x for different values of A and B. Solution Given equation is y = Ae2x+Be-2x dy dx = 2Ae2x - 2Be-2x Differentiate with respect to ‘x’ Again differentiate with respect to ‘x’ d 𝟐 y dx 𝟐 = 4Ae2x +4Be-2x Eliminate A and B from the above equations. We obtain the differential equation d 𝟐 y dx 𝟐 = 4y
- 19. DIFFERENTIAL EQUATIONS Solution ofa differential equation Definition: A relation between the variables without derivatives of a differential equation is said to be a solution or integral of the differential equation if the derivatives obtained there from, the equation is satisfied.
- 20. DIFFERENTIAL EQUATIONS Definition: A relation(x,y,c1,c2,…cn)=0 where c1,c2,…..,cn are n arbitrary constants is said to be the general solution or complete integral of the differential equation F(x,y,y1,y2….yn)=0. if (x,y,c1,c2,..cn) = 0 is a solution of F(x,y,y1,y2….yn) = 0
- 21. DIFFERENTIAL EQUATIONS NOTE: The numberof arbitrary constants in the general solution of a differential equation is equal to the order of the differential equation.
- 22. DIFFERENTIAL EQUATIONS PARTICULAR SOLUTION: If (x,y,c1,c2,…,cn) = 0 is the general solution of a differential equation F(x,y,y1,y2,…yn) = 0 Then (x,y,k1,k2…kn)=0, where k1,k2…kn are fixed constants is called a particular solution of the differential equation F(x,y,y1,y2…yn) = 0
- 23. DIFFERENTIAL EQUATIONS Singular solution: Asolution of a differential equation is said to be a singular solution if it is not a particular solution i.e., it cannot be obtained from the general solution by specifying values of arbitrary constants.
- 24. DIFFERENTIAL EQUATIONS Example: Considerthe differential equation d 𝟐 y dx 𝟐 +4y = 0 Solution Let y = sin2x Then dy dx = 2cos2x d 𝟐 y dx 𝟐+4y = 0 y = sin2x is a solution of d 𝟐 y dx 𝟐+4y = 0 y = cos2x is a solution of d 𝟐 y dx 𝟐+4y = 0 d2y dx2 = -4sin2x
- 25. DIFFERENTIAL EQUATIONS Further y= Acos2x+Bsin2x where A,B are arbitrary constants, is also a solution of d 𝟐 y dx 𝟐 +4y = 0 This solution is the general solution of d 𝟐 y dx 𝟐 +4y = 0 By a particular values to A,B we get particular solution of d 𝟐 y dx 𝟐 +4y = 0
- 26. DIFFERENTIAL EQUATIONS 1) Thenumber of arbitrary constants in the solution of a differential equation is equal to_______ 1) order 2) degree 3) number of terms in equation 4) number of differential coefficients
- 27. DIFFERENTIAL EQUATIONS 2) Theorder of the differential equation corresponding to y= A𝒆𝒙 +B𝒆𝟑𝒙 +C𝒆𝟓𝒙 (A,B,C being parameters) is ….. 1) 4 2) 1 3) 2 4) 3
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