1. This document contains 20 multiple part questions about differential equations. The questions cover topics like determining the degree and order of differential equations, solving differential equations, identifying whether equations are homogeneous, and forming differential equations to represent families of curves with given properties. 2. The questions range from 1 to 6 marks and include both conceptual questions about differential equations as well as problems requiring solving specific equations. A variety of solution techniques are required including separating variables, homogeneous property, and identifying particular solutions given initial conditions. 3. The document tests mastery of fundamental differential equation concepts and skills like classification, solving, identifying homogeneous property, and setting up equations to model geometric situations. A solid understanding of differential equations is needed to successfully answer all

1. CHAPTER – 9
DIFFERENTIAL EQUATION
1 Marks Questions:
Q.1. What is the degree and order of following differential equation?
(i) y
𝑑2 𝑦
𝑑𝑥2 +
𝑑𝑦
𝑑𝑥
3
= 𝑥
𝑑3 𝑦
𝑑𝑥3
2
. (ii)
𝑑𝑦
𝑑𝑥
4
+ 3y
𝑑2 𝑦
𝑑𝑥2 = 0. (iii)
𝑑3 𝑦
𝑑𝑥3 +y2
+ 𝑒
𝑑𝑦
𝑑 𝑥 = 0
(iv) y = x
𝑑𝑦
𝑑𝑥
+ a 1 +
𝑑𝑦
𝑑𝑥
2
(v) ym
– 3 (yn
)3
+ log y = 5x.
4 Marks Questions:
2. Solve the following differential equations :
(i)
𝑑𝑦
𝑑𝑥
+ y = cos x – sin x. (ii)
𝑑𝑦
𝑑𝑥
-
𝑦
𝑥
+ 𝑐𝑜𝑠𝑒𝑐
𝑦
𝑥
= 0; y = 0, x = 1.
(iii) (1 + y + x2
y)dx + (x + x3
)dy = 0; y = 0, x = 1. (iv) y ex/y
dx = (x ex/y
+ y)dy
(v)
𝑑𝑦
𝑑𝑥
+ y sec2
x = tan x sec2
x; y(0) = 1. (vi) (x2
+ 1)
𝑑𝑦
𝑑𝑥
+ 2xy = 𝑥2 + 4
(vii)
𝑑𝑦
𝑑𝑥
+ 2y tan x = sin x (viii)
𝑑𝑦
𝑑𝑥
= 𝑦 𝑡𝑎𝑛 𝑥, 𝑦 = 1 𝑥 = 0
Q.3. Form the differential equation representing the family of ellipse having foci on x – axis and centre at
origin.
Q.4. Show that the differential equation : 2y ex/y
dx + (y – 2x ex/y
) dy = 0 is homogenous and find its
particular solution, given x = 0 when y = 1.
Q.5. Form the differential equation representing the parabolas having vertex at origin and axis along
positive direction of x – axis.
Q.6. Solve the differential equations : (x + 𝑥𝑦) dy = y dx.
Q.7. Find the particular solution of the differential equation
𝑑𝑦
𝑑𝑥
+ y cot x = 4x cosec x : y = 0 when x =𝜋/2
Q.8. Form the differential equation of the family of circles having centre on y – axis and radius 3 units.
Q.9. Solve the following differential equations :
(i) (x2
– 1)
𝑑𝑦
𝑑𝑥
+ 2x y =
2
𝑥2−1
. (ii) x2
dy + y(x + y) dx= 0; y(1) = 1.
(iii) cos x
𝑑𝑦
𝑑𝑥
+ 𝑦 = 𝑠𝑖𝑛 𝑥. (iv) x ey/x
– y sin y/x + x
𝑑𝑦
𝑑𝑥
sin
𝑦
𝑥
= 0, y(1) = 0.
(v) (1 + y2
) dx = (tan-1
y – x) dy. (vi) x2 𝑑𝑦
𝑑𝑥
+ (1 – 2x) y = x2
.
(vii) xy log
𝑥
𝑦
𝑑𝑥 + 𝑦2
− 𝑥2
𝑙𝑜𝑔
𝑥
𝑦
dy. (viii) x log x
𝑑𝑦
𝑑𝑥
+ y = 2 log x.
(ix) 2xy + y2
– 2x2 𝑑𝑦
𝑑𝑥
= 0, y(1) = 2. (x)
𝑑𝑦
𝑑𝑥
+ y cot x = 2x + x2
cot x, y = 0 when x = 𝜋/2 .
(xi) x
𝑑𝑦
𝑑𝑥
= y – x tan
𝑦
𝑥
(xii) (x + 2y2
)
𝑑𝑦
𝑑𝑥
= y, given when x = 2, y = 1.
(xiii) x2
y dx – (x3
+ y3
) dy = 0. (xiv) (1 + ex/y
) dx + ex/y
1 −
𝑥
𝑦
dy = 0.
(xv) (3xy + y2
) dx + (x2
+ xy) dy = 0. (xvi) x dy – y dx = 𝑥2 + 𝑦2 dx

2. (xvii) x log x
𝑑𝑦
𝑑𝑥
+ y =
2
𝑥
log x.
Q.10. Form the differential equation representing the family of curves given by (x – a)2
+ 2y2
= a2
, where a is an
arbitracy constant.
Q.11. Solve : (1 + x2
)
𝑑𝑦
𝑑𝑥
+ y = tan-1
x.
Q.12. Form the differential equation of the family of circles in the second quadrant and touching the
coordinate axes.
Q.13. Form the differential equation representing the family of curves y2
– 2ay + x2
= a2
, where a is an
arbitrary constant.
Q.14. Find the particular solution of the following differential equation :
𝑑𝑦
𝑑𝑥
= 1 + 𝑥2
+ 𝑦2
+ 𝑥2
𝑦2
, given that y = 1 when x = 0.
Q.15. Solve : 1 + 𝑒
𝑥
𝑦 𝑑𝑥 + 𝑒
𝑥
𝑦 1 −
𝑥
𝑦
𝑑𝑦 = 0 .
Q.16. Solve the differential equation : (tan-1
y – x) dy = (1 + y2
) dx .
Q.17. Solve the following differential equation : cos2
x
𝑑𝑦
𝑑𝑥
+ y = tan x .
Q.18. Solve the following differential equation : 1 + 𝑥2 + 𝑦2 + 𝑥2 𝑦2 + 𝑥𝑦
𝑑𝑦
𝑑𝑥
= 0 .
Q.19. Show that the following differential equation is homogenous and then solve it.
Y dx + x log
𝑦
𝑥
𝑑𝑦 - 2x dy = 0.
Q.20. Show that the differential equation (x – y)
𝑑𝑦
𝑑𝑥
= x + 2y, is homogeneous and solve it.
6 Marks Questions :
Q.12. Solve the differential equations : (x 𝑥2 + 𝑦2 - y2
) dx + xy dy = 0.
Q.13. Find the particular solution of the differential equation : (x dy – y dx) y sin
𝑦
𝑥
= (ydx + xdy) x cosy/x,
Given y = 𝜋 when x = 3.
Q.14. Solve the equation : 𝑥 𝑐𝑜𝑠
𝑦
𝑥
+ 𝑦 𝑠𝑖𝑛
𝑦
𝑥
𝑦 𝑑𝑥 = 𝑦 𝑠𝑖𝑛
𝑦
𝑥
− 𝑥 𝑐𝑜𝑠
𝑦
𝑥
x dy.
Q.15. Show that the family of curves for which the slope of tangent at any point (x, y) on it is
𝑥2+𝑦2
2𝑥𝑦
, is given by
x2
– y2
= cx.
Q.16. Show that the differential equation (x – y)
𝑑𝑦
𝑑𝑥
= x + 2y is homogeneous and solve it.
Q.17. Find the general solution of the differential equation
𝑑𝑦
𝑑𝑥
- y = cos x.
Q.18. Find the particular solution of the differential equation :
(1 + e2x
) dy+ (1 + y2
) ex
dx = 0, given that y = 1 when x = 0.
Q.19. Solve : x dy – y dx = 𝑥2 + 𝑦2 dx
Q.20. Find the particular solution of the differential equation (1 + x3
)
𝑑𝑦
𝑑𝑥
+ 6x2
y = (1 + x2
), given that x = y = 1.