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.
