y'' + 2y' + y = 0 is a second-order equation and (y')² + y = 0 has degree 2, so this Class 12 sheet opens by counting orders and degrees. The key also answers y = x + c and y = x²/2 + c as general solutions.
Key Takeaways
y'' + 2y' + y = 0 has order 2 (Q1).
(y')² + y = 0 has degree 2 (Q2).
The key records y = x + c and y = x²/2 + c.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.The order of the differential equation y'' + 2y' + y = 0 is
(a) 1(b) 2(c) 3(d) 0
2.The degree of the differential equation (y')² + y = 0 is
(a) 1(b) 2(c) 3(d) 0
3.The general solution of dy/dx = 1 is
(a) y = x(b) y = x + c(c) y = x²/2 + c(d) y = c
4.The general solution of dy/dx = x is
(a) y = x²(b) y = x²/2 + c(c) y = x + c(d) y = x²/2
5.The general solution of dy/dx = 0 is
(a) y = x(b) y = c(c) y = cx(d) y = x + c
6.The order of the differential equation dy/dx + y = 0 is
(a) 0(b) 1(c) 2(d) 3
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Write the order of the differential equation y'' + 3y' = 0.
8.Write the degree of the differential equation (y')³ + y = 0.
9.Solve dy/dx = 2.
10.Solve dy/dx = x².
11.Write the general solution of dy/dx = 0.
12.Separate the variables in the equation dy/dx = xy.
13.Write the order and the degree of the differential equation dy/dx = sin x.
14.Solve dy/dx = eˣ.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.Solve dy/dx = 2x given that y = 5 when x = 1.
16.Find the general solution of dy/dx = 3x².
17.Solve the differential equation dy/dx = y given that y = 1 when x = 0.
18.The rate of growth of a population is proportional to the population. Write the differential equation representing this situation.
19.Solve dy/dx = 1/x for x > 0.
20.Find the particular solution of dy/dx = x + 1 if y = 0 when x = 0.
Answer Key
1. b) 2
2. b) 2
3. b) y = x + c
4. b) y = x²/2 + c
5. b) y = c
6. b) 1
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the order of y'' + 2y' + y = 0?
2, option (b) in this worksheet (Q1).
What is the degree of (y')² + y = 0?
2, option (b) in this worksheet (Q2).
What general solutions are recorded in the key?
y = x + c and y = x²/2 + c.
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