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Differential Equations Important Questions

12th Standard EM

Reg.No. :
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Time : 01:00:00 Hrs
Total Marks : 50
10 x 1 = 10
1. The differential equation ${ \left( \frac { dx }{ dy } \right) }^{ 3 }+2{ y }^{ \frac { 1 }{ 2 } }$ = x is

(a)

of order 2 and degree 1

(b)

of order 1 and degree 3

(c)

of order 1 and degree 6

(d)

of order 1 and degree 2

2. The integrating factor of the differential equation $\frac{dx}{dy}+Px=Q$

(a)

eഽPdx

(b)

eഽPdx

(c)

ഽPdy

(d)

eഽPdy

3. The complementary function of (D2+ 4)y = e2x is

(a)

(Ax +B)e2x

(b)

(Ax +B)e−2x

(c)

A cos 2x + B sin 2x

(d)

Ae−2x+ Be2x

4. The particular integral of the differential equation is $\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -8\frac { dy }{ dx }$+16y = 2e4x

(a)

$\frac { { x }^{ 2 }{ e }^{ 4x } }{ 2! }$

(b)

$\frac { { e }^{ 4x } }{ 2! }$

(c)

x2e4x

(d)

xe4x

5. Solution of $\frac { dy }{ dx }$ + Px = 0

(a)

x=cepy

(b)

x=ce−py

(c)

x = py + c

(d)

x = cy

6. The differential equation satisfied by all the straight lines in xy plane is

(a)

$\frac { dy }{ dx }$=a constant

(b)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$=0

(c)

y+ $\frac { dy }{ dx }$ = 0

(d)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$+y=0

7. If y = k.eλx then its differential equation where k is arbitrary constant is

(a)

$\frac { dy }{ dx }$=λy

(b)

$\frac { dy }{ dx }$=ky

(c)

$\frac { dy }{ dx }$+ky=0

(d)

$\frac { dy }{ dx }$=eλx

8. The differential equation obtained by eliminating a and b from y = a e3x + b e-3x is

(a)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$+ay=0

(b)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$-9y=0

(c)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } } -9\frac { dy }{ dx }$

(d)

$\frac { { d }^{ 2 }y }{ dx^{ 2 } }$+9x=0

9. The differential equation formed by eliminating A and B from y = ex (A cos x + B sin x) is

(a)

y2+y1=0

(b)

y2-y1=0

(c)

y2-2y1+2y=0

(d)

y2-2y1-2y=0

10. The degree of the differential equation $\sqrt { 1+\left( \frac { { d }y }{ dx } \right) ^{ \frac { 1 }{ 3 } } } =\frac { { d }^{ 2 }y }{ dx^{ 2 } }$ is

(a)

1

(b)

2

(c)

3

(d)

6

11. 5 x 1 = 5
12. y = mx+ c

13. (1)

$e^{ \int { pdy } }$

14. y= mx

15. (2)

Family of lines

16. Degree of linear differential equation

17. (3)

Family of parabolas having origin as vertex

18. General form of linear equation

19. (4)

1

20. I.F. of $\frac { dy }{ dx }$+Px=Q

21. (5)

$\frac { dy }{ dx }$+Py=Q

5 x 2 = 10
22. Find the differential equation of the family of all straight lines passing through the origin.

23. Form the differential equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x axis.

24. Find the differential equation of all circles passing through the origin and having their centers on the y axis.

25. Solve: ydx − xdy = 0

26. Solve: cosx(1 + cos y)dx − sin y(1 + sin x)dy = 0

27. 5 x 3 = 15
28. Find the differential equation of the family of curves $y=\frac { a }{ x } +b$ where a and b are arbitrary constants

29. Solve the differential equation $\frac { dy }{ dx } =\frac { x-y }{ x+y }$

30. Solve: (3D2 + D - 14)y = 4 - 13${ e }^{\frac{-7}{3}x}$

31. Suppose that the quantity demanded ${ Q }_{ d }=29-2p-5\frac { dp }{ dt } +\frac { { d }^{ 2 }p }{ { dt }^{ 2 } }$ and quantity supplied Qs= 5 + 4p where p is the price. Find the equilibrium price for market clearance.

32. Find the differential equation of all circles x2 +y2 + 2gx = 0 which pass through the origin and whose centres are on the X-axis.

33. 2 x 5 = 10
34. The marginal cost function of manufacturing x gloves is 6 +10x−6x2. The total  cost of producing a pair of gloves is Rs. 100. Find the total and average cost function.

35. The net profit p and quantity x satisfy the differential equation $\frac { dp }{ dx } =\frac { 2{ p }^{ 3 }-{ x }^{ 3 } }{ 3x{ p }^{ 2 } }$. Find the relationship between the net profit and demand given that p = 20, when x = 10.