Differential Equation
Sheet - 1
1. Form the differential equation in each of the following cases by eliminating the arbitrary
constants.
(i) x2 + y2 = 2ax
2.
(ii) x2-y2 = a2 (iii) y2 = 4ax (iv) y2 = m(a2 - x2)
y = aebx, where a, b are arbitrary constants.
3. y = a sin(x + b), where a and b are arbitrary constants.
4.
xy =Aex + Be~x + x2, where A, B are arbitrary constants.
5.
y2 = a(b - x)(b + x), where a, b are arbitrary constants.
6.
y = a sin(bx + c), where a, b, c are arbitrary constants.
7.
y2 - 2ay + x2 = a2, where a is an arbitrary constant.
8.
y = ex(a cos x + b sin x), where a, b are arbitrary constants.
9. Form the differential equation corresponding to (x — a)2 + (y — b)2 = r2 by eliminating a and b.
10. Form the differential equation corresponding to a (y + a)2 = x3 by eliminating a.
11. Form the differential equation corresponding to ax2 + by2 = 1 by eliminating a and b
12. Find the differential equation that will represent the family of the parallel lines y = 2x + c, where
c € R is a parameter.
13. Find the differential equation that will represent the family of straight lines y = mx + c, where m,
c <= R are parameters.
14. Determine the differential equation that will represent the family of all circles having centres on
the x-axis and radius of 2 units.
15. Find the differential equation of all circles in first quadrant which touch the coordinate axes.
16. Find the differential equation of all circles passing through the origin and having their centres on
(i) x-axis, (2005)
(ii) y-axis.
17. Find the differential equation of the family of all ellipses whose axes coincide with the coordinate
axes and centres are at the origin.
18. Find the differential equation of the family at parabolas whose (i) axis be parallel to y-axis, (ii)
Foci be at the origin and axis be along the x-axis.
19. Find the differential equation of the family of all parabolas with latus rectum 4a and whose axes
are parallel to x-axis.
20. Verify that y = A cos x - B sin x is a solution of the differential equation y’’ + y = 0 .
21. Show that y = cos ( cos x ) is a solution of the differential equation of y’’ – cot x y’ + y sin 2 x = 0
22. Show that y = b ex + c e2x is a solution of y’’ – 3 y’ + 2 y = 0 .
23. A spherical rain drop evaporates at a rate proportional to its surface area. Form a differential
equation involving the rate of change of the radius of the rain drop .