Mathematics_Today_-_October_2016

(iii) Reducible to variable separable : If the equation

is

dy

dx

=++fax by c(),

then put ax + by + c = z.

(iv) Homogeneous equation : If a first order, first
degree differential equation is expressible in the
form dy
dx

fxy

gxy

= (, )

(, )

, where f(x, y) and g(x, y) are

homogeneous functions of the same degree in x

and y, then put y = vx.

(v) Linear equation : If the equation is dy
dx

+=Py Q,

where P and Q are functions of x, then

(^) ye⋅ ∫Pdx=∫Qe⋅∫Pdxdx C+ ,^ where (^) e∫Pdx (^) is the
integrating factor (I.F.).
OR
If the equation is
dx
dy
+=Px Q, where P and Q are
functions of y, then
xe⋅∫Pdy=∫Qe⋅∫Pdydy C+ , where e∫Pdy is the
integrating factor (I.F.).
PROBLEMS
Very Short Answer Type

1. Solve the differential equation dy
   dx

=1.−xyxy+ −

2. Determine the order and degree of the differential

equation

dy

dx

dy

dx

2

2

2

=+ 1 ⎛⎝⎜ ⎞⎠⎟.

3. Find the differential equation of the family of all
   straight lines.


4. Find the integrating factor of the differential

equation e

x

y

x

dx

dy

x

− x

−

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

= ≠0).

2

1(

5. Find the order and degree of the differential
   equationypx ap b p

dy

dx

=+^22 +^2 ,.where =

Short Answer Type

6. Obtain the differential equation of the family of
   curves represented by y = Aex + Be–x + x^2 , where A
   and B are arbitrary constants.


7. Solve the differential equation log dy
   dx

⎛⎝⎜ ⎞⎠⎟=+ax by.

8. Solve : (1 + xy)ydx + (1 – xy)xdy = 0
   9. Verify that y = ae^3 x + be–x is a solution of the
   differential equation dy
   dx

dy

dx

y

2

2 −−^230 =

10. Solve the following differential equation
    dy
    dx

y

x

+=exx;0>

Long Answer Type-I

11. Solve the differential equation
    (1 + e^2 x)dy + ex(1 + y^2 )dx = 0. Given that y = 1,
    when x = 0.


12. Solve the following differential equation
    dy
    dx

+=2sinyx

13. Find the differential equation of the family of all
    circles touching the x-axis at the origin.


14. Show that the curve for which the normal at every
    point passes through a fixed point is a circle.


15. Solve the following differential equation
    dy
    dx

+secxy⋅ =tanx⎜⎛⎝ 0 ≤x< ⎞⎠⎟

2

π

Long Answer Type-II

16. Solve the following differential equation
    (x^3 + y^3 )dy – x^2 ydx = 0


17. In a bank, principal increases at the rate of 5% per
    year. In how many years Rs. 1000 double itself?


18. Find the general solution of the following differential
    equation y dx–(x + 2y^2 )dy = 0.


19. Solve the following differential equation

22012

xy y^22 x dy

dx

+ − ==;()y.

20. Solve the following differential equation
    dy
    dx

+= +yxx xxcot^2 cot 2

Also find the particular solution, given that y = 0

when x=π

2

.

SOLUTIONS

1. We h a v e dy
   dx

=()()11−xy+ ⇒ + = −

dy

y

xdx

1

() 1

⇒ ∫ + =∫ −⇒+=− +

dy

y

xdx y x x C

1

11

2

2

() log| |

2. We h a v e
   dy
   dx

dy

dx

2

2

2

=+ 1 ⎛⎝⎜ ⎞⎠⎟