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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 : 40
5 x 1 = 5
1. The differential equation of the family of curves y = Aex + Be−x, where A and B are arbitrary constants is

(a)

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

(b)

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

(c)

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

(d)

$\frac { { d }y }{ { dx } } -y=0$

2. The solution of the differential equation 2x$\frac{dy}{dx}-y=3$represents

(a)

straight lines

(b)

circles

(c)

parabola

(d)

ellipse

3. The integrating factor of the differential equation $\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda }$ is

(a)

$\frac { x }{ { e }^{ \lambda } }$

(b)

$\frac { { e }^{ \lambda } }{ x }$

(c)

${ \lambda e }^{ x }$

(d)

ex

4. The solution of (x2-ay)dx=(ax-y2)dy is

(a)

y=x2+y2-a(x+y)

(b)

y=x2+y2-a(x+y)

(c)

x3+y2=3ayx+c

(d)

(x2-ay)(ax-y2)=0

5. The transformation y=vx reduces $\\ \\ \\ \frac { dy }{ dx } =\frac { x+y }{ 3x }$

(a)

$\frac { 3av }{ 4v+1 } =\frac { dx }{ x }$

(b)

$\frac { 3dv }{ v+1 } =\frac { dx }{ x }$

(c)

$2x\frac { dv }{ dx } =v$

(d)

$\frac { 3dv }{ 1-2v } ==\frac { dx }{ x }$

6. 5 x 2 = 10
7. A differential equation, determine its order, degree (if exists)
$\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0$

8. A differential equation, determine its order, degree (if exists)
$y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } }$

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

10. Solve: x$\frac{dy}{dx}=x+y$

11. Solve: $\frac{dy}{dx}+y=e^{-x}$

12. 5 x 3 = 15
13. Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0

14. Show that y = 2(x2−1)+Ce−x2 is a solution of the differential equation $\frac { dy }{ dx } +2xy-4{ x }^{ 3 }=0$

15. Solve $(1+{ 2e }^{ x/y })dx+2{ e }^{ x/y }\left( 1-\frac { x }{ y } \right) dy=0$

16. Form the differential equation for y=e-2x [A coos 3x-B sin 3x]

17. Solve: $\frac{dy}{dx}+y=cos x$

18. 2 x 5 = 10
19. Solve y ' = sin2 (x − y + )1.

20. Solve:$\frac { dy }{ dx }$ =(3x+y+4)2.