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#### Ordinary Differential Equations Three Marks Questions Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 45
15 x 3 = 45
1. Determine the order and degree (if exists) of the following differential equations:
$\frac { dy }{ dx } =x+y+5$

2. Determine the order and degree (if exists) of the following differential equations:
${ \left( \frac { { d }^{ 4 }y }{ { dx }^{ 4 } } \right) }^{ 3 }+4{ \left( \frac { dy }{ dx } \right) }^{ 7 }+6y=5cos3x$

3. Determine the order and degree (if exists) of the following differential equations:
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right)$

4. Determine the order and degree (if exists) of the following differential equations:
$3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }$

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

6. Show that x2+y2 =r2, where r is a constant, is a solution of the differential equation $\frac { dy }{ dx }$ =-$\frac { x }{ y }$.

7. Show that y=mx+$\frac{7}{m}$,m≠0 is a solution of the differential equation xy'+7$\frac{1}{y'}$-y=0.

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

9. Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x2 y"+ xy'+y= 0.

10. Solve $\left( y+\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \right) dx-xdy=0,\quad y(1)=0$

11. Solve (2x + 3y)dx + ( y − x)dy = 0.

12. Solve ${ y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } =xy\frac { dy }{ dx }$

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

14. A radioactive isotope has an initial mass 200mg, which two years later is 50mg . Find the expression for the amount of the isotope remaining at any time. What is its half-life? (half-life means the time taken for the radioactivity of a specified isotope to fall to half its original value).

15. In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature and found it to be 70oF. Two hours later, the detective measured the body temperature again and found it to be 60oF. If the room temperature is 50oF, and assuming that the body temperature of the person before death was 98.6oF, at what time did the murder occur?
[log(2.43)=0.88789; log(0.5)=-0.69315]