Differential equations

Transcript

1 Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential equation: The order of a differential equation is the order of the highest deriatie occurring in it. Degree of differential equation: Degree of a differential equation is the degree of the highest deriatie occurring in it when the deriaties are made free from the radical sign. Solutions of differential equations of the first order and first degree: Variables separable method. Homogeneous equations. Non-Homogeneous equations: Linear equation: Equation reducible to linear form: Variable separable method. To sole dy XY,where X is a function of only dx and Y is a function of y only. Bring all the terms of and on one side, the terms of y and on the other side. Integrate both sides and add an arbitrary constant on one side.. Sole (y + )+ ( y + y) 0. Sol: (y + )+ ( y + y) 0. (y + )+ ( + ) y 0. by (y + )( + ) + y 0 ( +) (y +) ( + ) + y (y + ) 0 log( + ) + log( + y ) log c log( + )( + y ) log c ( + )( + y ) c 3. Sole + y +y Sol: y +y+ ++ y +y+ ++ y +y+( ) ( ) + (y+ ) ( 3 ) + ++( ) ( ) + (+ ) ( 3 ). Sole e y + e y Sol: e y + e y e ey + e y e + e y e y (e + ) +a a tan ( a ) + c 3 tan ( y + 3 ) + 3 tan ( + 3 ) c tan ( y + 3 ) + tan ( + 3 ) c e y (e + ) e y e + + c Saq e y Q.No e c

2 4. Sole (e + )y + (y + ) 0 Sol: (e + )y + (y + ) 0 by (e + )(y + ) (e + )y (e + )(y + ) + (y + ) (e + )(y + ) 0 y + (y+) 0 (e +) y+ (y+) + ( e +) 0 ( y+ ) + (y+) (y+) e 0 (+e ) (y+) e 0 (+e ) y log(y + ) log( + e ) log c y log(y + ) + log( + e ) + log c 6. Sole (log+) siny+ycosy Sol: (log+) siny+ycosy (siny + ycosy) (log + ) siny + y. cosy. log + cosy + y cosy [ cosy ] {log [ d (log ) ] } cosy + ysiny + cosy { log } + + c cosy + ysiny + cosy log + + c ysiny log + c y log(y + )( + e )c e y c(y + )( + e ) or e y e c(y + )( + e ) e +y c(y + )( + e ) 5. Sole y 5(y + ) Sol: y 5y y 5y (+5) y( 5y) 7. Sole + y 0 Sol: + y 0 Integrating on both sides + y 0 { a a + a sin ( a )} + sin () + y + sin (y) c + sin + y + sin y c y( 5y) (+5) Saq Q.No 7

3 8. Sole sin ( ) + y 0. Sole tan ( + y). Sol: sin ( ) + y sin( + y) put + y t + dt dt dt sin t dt + sin t +sin t sin t sin t sin t cos t dt dt dt [ cos t sint cost ] dt {sec t sec t tan t} dt tan t sec t + c 9. Sole tan( y) Sol: put y t dt The gien eq n becomes + dt tan t dt tan t dt. tan t cot dt log sint + c log sin (y ) + c Saq Q.No 7 3

4 Homogeneous differential equations To sole the equation f (,y) f (,y), where f (, y), f (, y)are homogeneous functions of the same degree in and y. Put y, so that + Substitute the alues of y and in the gien equation. Separate the ariables and. Integrate both sides and add an arbitrary constant on one side. Separate back the ariables y. + ( ) + ( ) ( ) ( )+ ( ) + ( ) ( ) + + ( ) ( ) ( ) ( ) + + ( ) ( ) ( ) ( ) f () log f() + c. f() log log log + log c log log c. Sole ( + y ) y.. Sol: ( + y ) y. y +y..() y ( y y ( c substituting y/ ) c y ) c let y + Eq n () + () +() ( + ) Saq Q.No 7 4

5 . Sole ( y ) y. 0. Sol: ( y ) y. 0 y y.. this is homogenous D. E let y + Eq n + () () + + ( ) + ( ) ( ) ( ) 4 4 ( f () log f() + c. f() 4 log log + logc log 4 log(c) ) 3. Sole ( y ) y. Sol: ( y ) y y y let y + Eq n () () ( 3 3) + ( ) log log + log c substituting y/ y y y log y/ log c log y/ +log c log y c y log yc + y log yc 0 log log(c) 4 ( ) (c) 4 ( ( y ) ) 4 c 4 y 4 c 4 ( y ) c 4 Saq Q.No 7 5

6 y y c 4. sole y +y Sol: this is homogenous D. E 5. sole ( y) (y ) sol: ( y) (y ) let y + y y this is homogenous D. E Eq n + +() + ( ) (+) + ( ) (+) ( ) (+) (( (+ (( (+ + (+) ) log log + log c log log c log log(c) (c) y (y ) c ) let y + Eq n + () + ( ) ( ) + ( ) ( ) ( ) ( ) ( + ) ( ) ( ) ( ) ( )(+) [ + ( ) 3 (+) ] ( ) 3 (+) log( ) 3 log( + ) log + log c log( ) log( + ) 3 log c log ( ) (+) 3 log c y y c Saq Q.No 7 ( ) (+) 3 c( y ) ( y + )3 c 6

7 (y ) 3 3 (y + )3 c (y ) (y + ) 3 c 6. sole (y + cos y ) Sol: (y + cos y ) y + cos ( y ) y + cos ( y ). this is homogenous D. E let y + Eq n + + cos () cos () cos () sec () tan log + c tan ( y ) log + c 7. Gien the solution of sin ( y ) y Which passes through the point (, π 4 ) Sol: sin ( y ) y y sin ( y ) [y sin ( y )] [y sin ( y )].. let y + + sin () sin () sin () cosec cot log + c cot ( y ) log + c this is passing Through the point (, π 4 ) cot ( π ) log + c c c cot ( y ) log 8. sole y+ +y 3. Sol: y+ +y 3 [a + b 0] this is non homogeneous D. E of case() Re grouping the terms properly ( + y 3) ( y + ) ( + ) (y 3) y 0 ( + ) (y 3) [y + ] 0 [ y sin ( y ) ] ( + ) (y 3) d(y) 0 Saq Q.No 7 7

8 + y 3y y c ( + + ) y y + 3y c 9. sole y+3 y+5. Sol: y+3 [ a b ] y+5 a b this is non homogeneous D. E of case() y+3.. ( y)+5 + log( + ) + c ( y) + log( y + ) + c y + log( y + ) c let ( y) Now eq n becomes (+)+ + ( (+) Saq Q.No ) 8

9 y. e tan (etan ) + c Linear differential equations To sole + Py Q, where P and Q are functions of only. Make the co-efficient of unity, if not so alrea. Find I. F e p and remember that e log f() f() the solution is y(i. F) Q(I. F) + c. Sole ( + ) + y 4 0 Sol: ( + ) + y y this is in the form of + Py Q where P + I. F e P, Q 4 + I. F e + e log(+) + the solution is y(i. F) Q(I. F) + c y( + ) 4 +. ( + ). Sole ( + ) + y etan Sol: ( + ) + y etan + y etan + + this is in the form of + Py Q where P, Q etan + + I. F e P I. F e + e tan the solution is y(i. F) Q(I. F) + c y. e tan e tan +. etan let e tan t etan dt y. e tan tdt + y( + ) 4. y( + ) c y( + ) c 3. Sole + y tan sin + y tan sin Sol: this is in the form of + Py Q where P tan, Q sin I. F e P I. F e tan e log sec I.Fsec the solution is y(i. F) Q(I. F) + c y(sec) sin sec y. e tan t Saq Q.No 7 y(sec) sin cos 9

10 y(sec) tan y(sec) log sec + c 4. Sole y tan e sec Sol: y tan e sec this is in the form of + Py Q where P tan, Q e sec I. F e P I. F e tan e log sec I.F sec cos the solution is y(i. F) Q(I. F) + c y(cos) e sec. cos y(cos) e y(cos) e + c 5. Sole ( + y + ) Sol: ( + y + ) +y+ - y + + y + this is in the form of + P Q where P, Q y + I. F e P I. F e e y I.Fe y the solution is (I. F) Q(I. F) + c (e y ) e y (y + ) (e y ) (y + )e y + e y (e y ) (y + )e y + e y + c (y + ) + ce y + y + ce y which is the required solution. 6. Sole ( + y ) (tan y ) Sol: ( + y ) (tan y ) tan y (+y ) + tan y (+y ) (+y ) this is in the form of + P Q where P ( + y ), Q tan y ( + y ) I. F e P I. F e (+y ) e tan y I.Fe tan y the solution is (I. F) Q(I. F) + c (e tan y ) e tan y. tan y (+y ) let tan y t dt (+y ) (e tan y ) e t. t. dt (e tan y ) e t (e t ) + c (e tan y ) e tan y (e tan y ) + c 7. Sole cos + y sin sec Sol: cos + y sin sec (e y ) (y + ) e y [ d (y + ) e y ] (e y ) (y + )e y. e y Saq Q.No 7 sin + y. sec cos cos + y. tan sec3 this is in the form of + Py Q 0

11 where P tan, Q sec 4 I. F e P I. F e tan e log sec I.Fsec the solution is y(i. F) Q(I. F) + c y(sec) sec 3. sec y(sec) sec 4 y(sec) ( + tan ) sec let tan t sec dt y(sec) ( + t )dt y(sec) t + t3 3 + c y(sec) tan + tan3 + c 3 8. Sole log + y log Saq Q.No 7