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### MB1209 QR

1. 1. In this document you can review important • Definitions • Facts • Procedures Class –XII, CBSE MB1209: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com
2. 2. What is Differential Equation? Class – XII,CBSE MB1209: Differential Equations Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com
3. 3. What is Differential Equation? Class – XII,CBSE Important Definitions An equation involving derivatives of the dependent variable with respect to independent variable (variables) is called a .differential equation Learn Math for Free. Visit www.ganitgurooz.com Topic: Differential EquationsMB1209: Differential Equations
4. 4. What is ordinary Differential Equation? Class –XII,CBSE Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Differential EquationsMB1209: Differential Equations
5. 5. What is ordinary Differential Equation? Class – XII,CBSE Important Definitions A differential equation involving derivative of the dependent variable with respect to only one independent variable is called an . ordinary differential equation Learn Math for Free. Visit www.ganitgurooz.com Topic: Differential EquationsMB1209: Differential Equations
6. 6. What is the order differential equation? Class – XII,CBSE Important Definitions Learn Math for Free. Visit www.ganitgurooz.com Topic: Differential EquationsMB1209: Differential Equations
7. 7. What is the order differential equation? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com of a differential equation is the order of the highest order derivative occurring in the differential equation. Order MB1209: Differential Equations
8. 8. What is the degree differential equation? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations
9. 9. What is the degree differential equation? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com of a differential equation is the highest power (positive integer only) of the highest order derivative in it. Degree MB1209: Differential Equations
10. 10. What is the solution of differential equation and its types? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations
11. 11. What is the solution of differential equation and its types? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com A function which satisfies the given differential equation is called its . The solution which contains as many arbitrary constants as the order of the differential equation is called a solution general and the solution free from arbitrary constants is called . solution particular solution MB1209: Differential Equations
12. 12. What is a homogeneous function? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations
13. 13. What is a homogeneous function? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com A function ( , ) is a homogeneous function of deg ree if, F( , ) orn n F x y n y x x y x g y h x y MB1209: Differential Equations
14. 14. What is a homogeneous differential equation? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations
15. 15. What is a homogeneous differential equation? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com A differential equation which can be expressed in the form ( , ) or ( , ) where, ( , ) and ( , ) are homogenous functions of degree zero is called a . dy f x y dx dx g x y f x y dy g x y homogeneous differential equation MB1209: Differential Equations
16. 16. What is a first order linear differential equation? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations
17. 17. What is a first order linear differential equation? Class – XII,CBSE Important Definitions Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com A differential equation of the form P Q, where P and Q are constants or functions of only is called a first order . dy x dx x linear differential equation MB1209: Differential Equations
18. 18. Fact # 1 Class – XII,CBSE Important Facts Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com of a differential equation is defined if it is a polynomial equation in its derivatives. Degree MB1209: Differential Equations
19. 19. Fact # 2 Class – XII,CBSE Important Facts Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com To form a differential equation from a given function we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants. MB1209: Differential Equations
20. 20. Fact # 3 Class – XII,CBSE Important Facts Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com Variable separable method is used to solve such an equation in which variables can be separated completely i.e. terms containing should remain with and terms containing should remain with . y dy x dx MB1209: Differential Equations
21. 21. How to form a differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations 1 1 : If the given family F of curves depend on only one parameter then it is represented by the equation of the form ( , , ) 0. Differentiate the above equation with respect to x a One parameter F x y a Step1 nd get an equation of the form ( , , , ) 0. The required differential equation is obtained by eliminating a from the equation as ( , , ) 0. g x y y a F x y y
22. 22. How to form a differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations 2 2 : If the given family F of curves depend on the parameters , (say) then it is represented by the equation of the form F ( , , , ) 0 ......(1) Differentiate equation (1) with resp Two parameter a b x y a b Step 2 ect to x, we get an equation of the form ( , , , , ) 0 ......(2) Differential equation (2) with respect to x to get an equation of the form ( , , , , , ) 0 ......(3) g x y y a b h x y y y a b
23. 23. How to form a differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations The required differential equation is obtained by eliminating and from the equation (1),(2) and (3) as ( , , , ) 0. a b F x y y y
24. 24. How to solve first order, first degree differential equation by variable separable method? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations
25. 25. How to solve first order, first degree differential equation by variable separable method? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations : A first order first degree differential equation is of the form, F( , ) ...... 1 Express ( , ) as a product ( ) ( ), where ( ) is a function of and ( ) is a function of . The different dy x y dx F x y g x h y g x x h y y Step 1 ial equation (1) then has the form: ( ) ( ) ...... 2 dy h y g x dx
26. 26. How to solve first order, first degree differential equation by variable separable method? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations If ( ) 0, separating the variables, (2) can be rewritten as 1 ( ) ...... 3 ( ) Integrate both sides of (3), we get 1 ( ) ...... 4 ( ) h y dy g x dx h y dy g x dx h y
27. 27. How to solve first order, first degree differential equation by variable separable method? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations Thus, the solution of the given differential equation is of the form ( ) ( ) , where ( ) and ( ) are the anti derivatives if 1 and ( ) respectively and is the arbitrary constant. ( ) H y G x C H y G x g x C h y
28. 28. How to solve first order, first degree homogeneous differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations : The differential equation is of the type F( , ) ...... 1 Substitute ...... 2 dy y x y g dx x y v x Step1
29. 29. How to solve first order, first degree homogeneous differential equation? Class – XII,CBSE Important Procedures Topic: Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations Differentiate equation (2) with respect to , we get ...... 3 Substitute the value of from equation (3) in equation (1), and get ( ) ...... 4 x dy dv v x dx dx dy dx dv x g v v dx
30. 30. How to solve first order, first degree homogeneous differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations Separating the variables in equation (4), we get ...... 5 ( ) Integrating both sides of equation (5) we get ...... 6 ( ) dv dx g v v x dv dx C g v v x
31. 31. How to solve first order, first degree homogeneous differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations The general solution of the differential equation (1) is given by equation (6) when we replace by . y v x
32. 32. How to solve first order, first degree homogeneous differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations If the homogeneous differential is of the form F( , ), where ( , ) is a homogeneous function of degree zero , then we make the substitution and proceed further to find the general solutio dx x y F x y dy x v y n as discussed above by writing F( , ) h . dx x x y dy y
33. 33. How to solve first order linear differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations P : The given differential equation in the form P Q where P and Q are constants or functions of only. Find the integrating factor (I.F) = e . Write the solution of the given differenti dx dy y x dx Step1 al equation as I.F (Q I.F) .y dx C
34. 34. How to solve first order, first degree linear differential equation? Class – XII,CBSE Important Procedures Topic: Differential Equations Learn Math for Free. Visit www.ganitgurooz.com MB1209: Differential Equations 1 1 1 1 1 P : The given differential equation is of the form P Q where P and Q are constants or functions of only. Find the integrating factor (I.F) = e . Write the solution of the given diff dy dx x y dy Step 2 1 erential equation as . I.F (Q I.F) .x dy C