Download Differential Equations - Mathematics and more Study notes Mathematics in PDF only on Docsity!
9.1 Overview
(i) An equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation.
(ii) A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation and a differential equation involving derivatives with respect to more than one independent variables is called a partial differential equation.
(iii) Order of a differential equation is the order of the highest order derivative occurring in the differential equation.
(iv) Degree of a differential equation is defined if it is a polynomial equation in its derivatives.
(v) Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it.
(vi) A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
(vii) 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.
(viii) The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves.
(ix) ‘Variable separable method’ is used to solve such an equation in which variables can be separated completely, i.e., terms containing x should remain with dx and terms containing y should remain with dy.
Chapter 9
DIFFERENTIAL EQUATIONS
180 MATHEMATICS
(x) A function F ( x , y ) is said to be a homogeneous function of degree n if F (λ x, λ y )= λ n^ F ( x , y ) for some non-zero constant λ.
(xi) A differential equation which can be expressed in the form
dy dx = F ( x ,^ y ) or dx dy = G ( x ,^ y ), where F ( x ,^ y ) and G ( x ,^ y ) are homogeneous functions of degree zero, is called a homogeneous differential equation_._
(xii) To solve a homogeneous differential equation of the type
dy dx = F ( x ,^ y ), we make substitution y = vx and to solve a homogeneous differential equation of the type dx dy = G ( x ,^ y ), we make substitution^ x^ =^ vy.
(xiii) A differential equation of the form
dy dx + P y^ = Q, where P and Q are constants or functions of x only is known as a first order linear differential equation. Solution of such a differential equation is given by y (I.F.) = (^) ∫ ( Q^ ×I.F.)^ dx + C, where
I.F. (Integrating Factor) = (^) e ∫^ P dx.
(xiv) Another form of first order linear differential equation is
dx dy + P^1 x^ = Q^1 , where P 1 and Q 1 are constants or functions of y only. Solution of such a differential equation is given by x (I.F.) = (^) ∫ ( Q × I.F. 1 )^ dy + C, where I.F. = (^) e ∫ P 1 dy.
9.2 Solved Examples
Short Answer (S.A.)
Example 1 Find the differential equation of the family of curves y = A e^2 x^ + B. e –2 x.
Solution y = A e^2 x^ + B. e –2 x
182 MATHEMATICS
y.x = (^) ∫ x x^2 dx , i.e. yx =
4 4
x (^) + c
Hence y =
3 4
x c x
Example 5 Find the differential equation of the family of lines through the origin.
Solution Let y = mx be the family of lines through origin. Therefore,
dy dx =^ m
Eliminating m , we get y =
dy dx.^ x^ or^ x^
dy dx –^ y^ = 0. Example 6 Find the differential equation of all non-horizontal lines in a plane. Solution The general equation of all non-horizontal lines in a plane is ax + by = c , where a ≠ 0.
Therefore, a dx b dy
Again, differentiating both sides w.r.t. y , we get 2 2 a^ d x dy = 0^ ⇒
2 2
d x dy = 0.
Example 7 Find the equation of a curve whose tangent at any point on it, different
from origin, has slope
y y x
Solution Given
dy y y dx x
y 1 x
⇒^ dy^ 1 1 dx y x
= ^ +
Integrating both sides, we get
log y = x + log x + c ⇒ log y x
=^ x^ +^ c
DIFFERENTIAL EQUATIONS 183
y x =^ e
x + c (^) = ex.ec (^) ⇒ y x =^ k. e
x
⇒ y = kx. ex.
Long Answer (L.A.)
Example 8 Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P( x , y ) of the curve is equal to the distance of P from the x – axis.
Solution Let the equation of normal at P( x , y ) be Y – y = (^ )
Y + X^
dx dy –^ y x dx dy
= 0^ ...(1)
Therefore, the length of perpendicular from origin to (1) is
2 1
y x^ dx dy dx dy
+ ^
Also distance between P and x -axis is | y |. Thus, we get
2 1
y x^ dx dy dx dy
= | y |
dx^2 y x dy
2 y^2 1 dx dy
^ ^ ⇒^ (^ )
dx dx (^) x (^2) – y (^22) xy 0 dy dy
⇒^ dx 0 dy
or dx dy
xy y x
DIFFERENTIAL EQUATIONS 185
⇒ sec^2 v dv = dx x − (^) ⇒ tan v = – logx + c
⇒ tan log
y (^) x c x
Substituting x = 1, y = (^4) π , we get. c = 1. Thus, we get
tan
y x
+ log^ x^ = 1, which is the required equation.
Example 10 Solve 2 x dy xy dx
− (^) = 1 + cos y x
, x ≠ 0 and x = 1, y = (^2)
π
Solution Given equation can be written as
x^2^ dy xy dx
− (^) = 2cos^2 2
y x
, x ≠ 0.
2
2
2cos 2
x dy xy dx y x
2 2
sec (^2 ) 2
y x (^) x dy xy dx
−^ =
Dividing both sides by x^3 , we get
2 2 3
sec 2 1 2
y (^) dy x x^ y dx x x
tan 1 2
d y dx x (^) x
Integrating both sides, we get
2 tan 1 (^2 )
y (^) k x (^) x
186 MATHEMATICS
Substituting x = 1, y = (^2)
π , we get
k =
2 , therefore,^2
tan 1 3 2 2 2
y x (^) x
is the required solution.
Example 11 State the type of the differential equation for the equation.
xdy – ydx = (^) x^2^ + y^2 dx and solve it.
Solution Given equation can be written as xdy = (^) (^ x^^2 +^ y^^2 +^ y dx ) , i.e.,
dy^ x^2^^ y^2^ y dx x
Clearly RHS of (1) is a homogeneous function of degree zero. Therefore, the given equation is a homogeneous differential equation. Substituting y = vx , we get from (1)
dv^ x^^2 v x^2^^2 vx v x dx x
x dv 1 v^2 dx
dv dx v x
Integrating both sides of (2), we get
log ( v + (^1) + v^2 ) = log x + log c ⇒ v + (^1) + v^2 = cx
y x +
2 (^1 ) y x
- (^) = cx ⇒ y + (^) x^2 + y^2 = cx^2
188 MATHEMATICS
Solution Correct answer is (C). Given equation can be written as
2 3
dy dx y x
- ⇒^ 2log ( y^ + 3) = log x^ + log c
⇒ ( y + 3)^2 = cx which represents the family of parabolas
Example 17 The integrating factor of the differential equation
dy dx ( x^ log^ x ) +^ y^ = 2log x^ is (A) ex^ (B) log x (C) log (log x ) (D) x
Solution Correct answer is (B). Given equation can be written as
log
dy y dx x x x
Therefore, I.F. =
1 e ∫^ x^ log^ xdx =^ e log (log x )^ = log^ x.
Example 18 A solution of the differential equation
2 dy (^) x dy y 0 dx dx
is
(A) y = 2 (B) y = 2 x (C) y = 2 x – 4 (D) y = 2 x^2 – 4
Solution Correct answer is (C).
Example 19 Which of the following is not a homogeneous function of x and y.
(A) x^2 + 2 xy (B) 2 x – y (C) cos^2 y^ y x x
(D) sin x – cos y
Solution Correct answer is (D).
Example 20 Solution of the differential equation 0
dx dy x y
(A)
(^1 1) c x y
- = (^) (B) log x. log y = c (C) xy = c (D) x + y = c
Solution Correct answer is (C). From the given equation, we get log x + log y = log c giving xy = c.
DIFFERENTIAL EQUATIONS 189
Example 21 The solution of the differential equation 2 2
dy x y x dx
(A)
2 4 2
y x^ c x
(B)
2 4 y =^ x + c (C)^4 2 y x^ c x
(D)
4 4 2
y x^ c x
=^ +
Solution Correct answer is (D). I.F. = 2
2 e^ ∫^ x^ dx^ = e 2log^ x^ = e log^ x = x^2. Therefore, the solution
is y. x^2 = 2 4 . 4
x xdx =^ x + k ∫ , i.e.,^ y^ =
4 42
x c x
Example 22 Fill in the blanks of the following:
(i) Order of the differential equation representing the family of parabolas y^2 = 4 ax is __________.
(ii) The degree of the differential equation
3 2 2 2 0
dy d y dx dx
^ ^
+^ ^ =
is ________.
(iii) The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is __________.
(iv) F ( x , y ) = x^2 y^2 y x
- (^) is a homogeneous function of degree__________.
(v) An appropriate substitution to solve the differential equation
dx dy
(^2) log 2
log
x x x y xy x y
is__________.
(vi) Integrating factor of the differential equation (^) x dy y dx
− = sin x^ is __________.
(vii) The general solution of the differential equation x^ y
dy (^) e dx
= − is __________.
DIFFERENTIAL EQUATIONS 191
dy dx =
e^ –2^ x y x x
− (^) i.e. dy dx +^
y x =
e^ –2^ x x
This is a differential equation of the type dy dx + P y^ = Q. Example 23 State whether the following statements are True or False. (i) Order of the differential equation representing the family of ellipses having centre at origin and foci on x -axis is two.
(ii) Degree of the differential equation
2 (^1 ) d y dx
- (^) = x + dy dx is not defined.
(iii) 5 dy (^) y dx
(iv) F( x , y ) =
cos
cos
y y x x x y x
is not a homogeneous function.
(v) F( x , y ) =
x^2^ y^2 x y
− is a homogeneous function of degree 1.
(vi) Integrating factor of the differential equation cos dy (^) y x dx − = (^) is ex.
(vii) The general solution of the differential equation x (1 + y^2 ) dx + y (1 + x^2 ) dy = 0 is (1 + x^2 ) (1 + y^2 ) = k.
(viii) The general solution of the differential equation sec dy (^) y x dx
- (^) = tan x is y (sec x – tan x ) = sec x – tan x + x + k.
(ix) x + y = tan–1 y is a solution of the differential equation y^2 2 1
dy (^) y dx
192 MATHEMATICS
(x) y = x is a particular solution of the differential equation
(^22) 2
d y (^) x dy xy x dx dx
Solution
(i) True, since the equation representing the given family is
2 2 2 2 1
x y a b
has two arbitrary constants.
(ii) True, because it is not a polynomial equation in its derivatives.
(iii) True
(iv) True, because f ( λ x , λ y ) = λ° f ( x , y ).
(v) True, because f ( λ x , λ y ) = λ^1 f ( x , y ).
(vi) False, because I.F = (^) e ∫^ −^1 dx^ = e – x.
(vii) True, because given equation can be written as
2 2
x (^) dx ydy x y
=^ −
⇒ log (1 + x^2 ) = – log (1 + y^2 ) + log k ⇒ (1 + x^2 ) (1 + y^2 ) = k
(viii) False, since I.F. = (^) e ∫^ sec^ xdx^ = e log(sec^ x^ +tan^ x )= sec x + tan x , the solution is,
y (sec x + tan x ) = (^) ∫ (sec^ x^ +tan^ x ) tan^ xdx = (^) ∫ ( sec^ x^ tan^ x^ + sec^2 x^ −^1 ) dx =
sec x + tan x – x +k
(ix) True, x + y = tan–1 y ⇒ (^1 ) 1
dy dy dx (^) y dx
dy dx y
, i.e.,
2 2
dy (1 y ) dx (^) y = −^ + which satisfies the given equation.
194 MATHEMATICS
13. Form the differential equation having y = (sin–1 x )^2 + Acos–1 x + B, where A and B are arbitrary constants, as its general solution. 14. Form the differential equation of all circles which pass through origin and whose centres lie on y -axis. 15. Find the equation of a curve passing through origin and satisfying the differential
equation (1^2 )^2 x dy xy x dx
16. Solve : x^2
dy dx =^ x
(^2) + xy + y (^2).
17. Find the general solution of the differential equation (1 + y^2 ) + ( x – e tan–1 y )
dy dx = 0.
18. Find the general solution of y^2 dx + ( x^2 – xy + y^2 ) dy = 0. 19. Solve : ( x + y ) ( dx – dy ) = dx + dy. [ Hint: Substitute x + y = z after seperating dx and dy ] 20. Solve : 2 ( y + 3) – xy
dy dx = 0, given that^ y^ (1) = – 2.
21. Solve the differential equation dy = cos x (2 – y cosec x ) dx given that y = 2 when
x = π.
22. Form the differential equation by eliminating A and B in A x^2 + B y^2 = 1. 23. Solve the differential equation (1 + y^2 ) tan–1 x dx + 2 y (1 + x^2 ) dy = 0. 24. Find the differential equation of system of concentric circles with centre (1, 2).
Long Answer (L.A.)
25. Solve : (^ ) y d xy dx
- (^) = x (sin x + log x ) 26. Find the general solution of (1 + tan y ) ( dx – dy ) + 2 xdy = 0. 27. Solve : dy dx = cos( x + y ) + sin ( x + y ).[ Hint: Substitute x + y = z ] 28. Find the general solution of 3 sin 2
dy (^) y x dx
29. Find the equation of a curve passing through (2, 1) if the slope of the tangent to
the curve at any point ( x , y ) is
2 2 2
x y xy
DIFFERENTIAL EQUATIONS 195
30. Find the equation of the curve through the point (1, 0) if the slope of the tangent
to the curve at any point ( x , y ) is (^2)
y 1 x x
31. Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point ( x , y ) is equal to the square of the difference of the abcissa and ordinate of the point. 32. Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P ( x , y ) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB. 33. Solve : x dy y dx
= (^) (log y – log x + 1)
Objective Type Choose the correct answer from the given four options in each of the Exercises from 34 to 75 (M.C.Q)
34. The degree of the differential equation
2 2 2 2 sin^ is:
d y dy (^) x dy dx dx dx
+^ ^ ^ = ^
(A) 1 (B) 2 (C) 3 (D) not defined
35. The degree of the differential equation
3 (^2 ) 1 2 is dy d y dx (^) dx
+^ =
(A) 4 (B)
2 (C) not defined^ (D) 2
36. The order and degree of the differential equation
2 1 1 (^4 ) 2 +^0
d y dy (^) x dx dx
+ ^ =
respectively, are (A) 2 and not defined (B) 2 and 2 (C) 2 and 3 (D) 3 and 3
37. If y = e – x^ (Acos x + Bsin x ), then y is a solution of
(A)
2 2 2 0
d y dy dx dx
+ = (B)
2 2 2 2 0
d y dy (^) y dx dx
(C)
2 2 2 2 0
d y dy (^) y dx dx
+ + = (D)
2 2 2 0
d y (^) y dx
DIFFERENTIAL EQUATIONS 197
45. The number of solutions of
dy y dx x
− when^ y^ (1) = 2 is : (A) none (B) one (C) two (D) infinite
46. Which of the following is a second order differential equation?
(A) ( y ′)^2 + x = y^2 (B) y ′ y ′′^ + y = sin x
(C) y ′′′^ + ( y ′′^ )^2 + y = 0 (D) y ′^ = y^2
47. Integrating factor of the differential equation (1 – x^2 ) 1 dy (^) xy dx − = (^) is
(A) – x (B) 1 2
x
- x (C)^ 1 − x^2 (D)^1 2 log (1 –^ x
48. tan–1^ x + tan–1^ y = c is the general solution of the differential equation:
(A)
2 2
dy y dx x
+ (B)
2 2
dy x dx y
(C) (1 + x^2 ) dy + (1 + y^2 ) dx = 0 (D) (1 + x^2 ) dx + (1 + y^2 ) dy = 0
49. The differential equation y
dy dx +^ x^ =^ c^ represents : (A) Family of hyperbolas (B) Family of parabolas (C) Family of ellipses (D) Family of circles
50. The general solution of ex^ cos y dx – ex^ sin y dy = 0 is : (A) ex^ cos y = k (B) ex^ sin y = k (C) ex^ = k cos y (D) ex^ = k sin y 51. The degree of the differential equation
2 3 5 2 6 0
d y dy (^) y dx dx
+ ^ + =
is : (A) 1 (B) 2 (C) 3 (D) 5
52. The solution of –^ x ,^ (0)^0
dy y e y dx
- = = (^) is : (A) y = ex^ ( x – 1) (B) y = xe–x (C) y = xe–x^ + 1 (D) y = ( x + 1) e–x
198 MATHEMATICS
53. Integrating factor of the differential equation tan^ – sec^0 dy (^) y x x dx
(A) cos x (B) sec x (C) e cos x^ (D) e sec x
54. The solution of the differential equation
2 2
dy y dx (^) x
- is: (A) y = tan–1 x (B) y – x = k (1 + xy ) (C) x = tan–1 y (D) tan ( xy ) = k
55. The integrating factor of the differential equation dy (^) y 1 y dx x
is:
(A) (^) x
x e (B)
e^ x x (C) xex^ (D) ex
56. y = aemx^ + be – mx^ satisfies which of the following differential equation?
(A) 0
dy (^) my dx
(C)
(^22) 2 0
d y (^) m y dx
− = (D)
(^22) 2 0
d y (^) m y dx
57. The solution of the differential equation cos x sin y dx + sin x cos y dy = 0 is :
(A)
sin sin
x (^) c y
= (^) (B) sin x sin y = c
(C) sin x + sin y = c (D) cos x cos y = c
58. The solution of x dy dx +^ y^ =^ e
x (^) is:
(A) y = e x k x x
(C) y = xex^ + k (D) x =
e y k y y
