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#### Ordinary Differential Equations Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. The general solution of the differential equation $\frac { dy }{ dx } =\frac { y }{ x }$ is

(a)

xy = k

(b)

y = k log x

(c)

y = kx

(d)

log y = kx

2. If p and q are the order and degree of the differential equation $y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,$When

(a)

p < q

(b)

p = q

(c)

p>q

(d)

p exists and q does not exist

3. The slope at any point of a curve y = f (x) is given by $\frac{dy}{dx}=3x^2$ and it passes through (-1,1). Then the equation of the curve is

(a)

y=x3+2

(b)

y=3x2+4

(c)

y=3x4+4

(d)

y=3x2+5

4. The I.F. of cosec x$\frac{dy}{dx}+y$sec2x=0 is

(a)

esec x

(b)

etan x

(c)

esec x tan x

(d)

esec2 x

5. The differential equation associated with the family of concentric circles having their centres at the origin is _________.

(a)

$\frac { dy }{ dx } =\frac { -x }{ y }$

(b)

$\frac { dy }{ dx } =\frac { -y }{ x }$

(c)

$\frac { dy }{ dx } =\frac { x }{ y }$

(d)

$\frac { dy }{ dx } =\frac { y }{ x }$

6. 2 x 2 = 4
7. If $\frac { dy }{ dx } =\frac { x-y }{ x+y }$ then
1) xdy+y dx=x dx+y dy
2) $\int { d(xy) } =\int { xdx } +\int { ydy }$
3) x2-y2+2xy=c
4) x2-y2-2xy=c

8. Solution of $\frac{dy}{dx}$+mx=0 where m<0 is
1) $\frac{dy}{dx}$=-m dy
2) y=cemx
3) log x=-my+log x
4) x=ce-my

9. 6 x 2 = 12
10. A differential equation, determine its order, degree (if exists)
$\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0$

11. A differential equation, determine its order, degree (if exists)
${ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0$

12. Find value of m so that the function y = emx is a solution of the given differential equation.
y '+ 2y = 0

13. Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0

14. Solve the differential equation:
$\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } }$

15. Solve: $\frac{dy}{dx}=1+e^{x-y}$

16. 3 x 3 = 9
17. Express the physical statement in the form of the differential equation.
For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.

18. Find the differential equation of the family of all non-vertical lines in a plane.

19. Solve: x$\frac{dy}{dx}$+2y=x2

20. 4 x 5 = 20
21. Solve the differential equation:
(ydx-xdy)cot$\left( \frac { x }{ y } \right)$=ny2 dx

22. Solve the differential equation:
tan y$\frac{dy}{dx}$=cos(x+y)+cos(x-y)

23. Solve the Linear differential equation:
$(1+x+{ xy }^{ 2 })\frac { dy }{ dx } +(y+{ y }^{ 3 })=0$

24. A population grows at the rate of 2% per year. How long does it take for the population to double?