Differential Equations

Maths, HSSC-Punjab Second Year. This chapter is split into 31 topics and has 733 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Constructing differential equations: inflation, deflation and growth of money 20 questions
2. Constructing differential equations: RL circuit 22 questions
3. Applications: Newton's law of cooling and heating 22 questions
4. Differential equations: finding the order 29 questions
5. Applications: RC circuit charge and current 25 questions
6. Constructing differential equations: RC circuit 20 questions
7. Differential equations: definition and basic terms 23 questions
8. Differential equations: finding the degree 30 questions
9. Separable equations: general solutions (trigonometric, inverse and logarithmic forms) 21 questions
10. Separable equations: general solutions (algebraic and exponential) 20 questions
11. Separable equations: recognising the equation from its solution 26 questions
12. Homogeneous equations: testing homogeneity 24 questions
13. Separable equations: initial value problems 30 questions
14. Separable differential equations: recognising and classifying 30 questions
15. Constructing differential equations: population growth and decay 20 questions
16. Differential equations: order and degree together 26 questions
17. Applications: growth constant and time for population models 20 questions
18. Constructing differential equations: Newton's law of cooling 22 questions
19. Applications: RL circuit current 23 questions
20. Homogeneous equations: initial value problems 26 questions
21. Applications: half-life 24 questions
22. Homogeneous equations: reducing to x dv/dx = F(v) - v 22 questions
23. Separable equations: power-function families dy/dx = x^m / y^n 24 questions
24. Homogeneous equations: the substitution y = vx 20 questions
25. Constructing differential equations: motion and other situations 20 questions
26. Homogeneous equations: general solutions 22 questions
27. Applications: choosing the model and reading its solution 20 questions
28. Applications: radioactive decay and mass remaining 20 questions
29. Separable differential equations: separating the variables 28 questions
30. Applications: bacterial growth from doubling and given data 26 questions
31. Applications: inflation, deflation and growth of money 28 questions

Sample questions: Differential Equations

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. What is a differential equation?

• A. An equation that contains no variable at all
• B. An equation containing a derivative of a dependent variable(correct answer)
• C. An equation containing only polynomials in one variable
• D. An equation containing only the integral of a function

Answer: B.The book defines a differential equation as an equation that involves at least one derivative of a dependent variable with respect to an independent variable.

Q2. 2x (d²y/dx²)⁴ + 2x + dy/dx = 0: what is the degree of this differential equation?

• A. 1
• B. 2
• C. 3
• D. 4(correct answer)

Answer: D.The highest-order derivative is of order 2 and it appears raised to the power 4, so the degree is 4. (Powers of lower derivatives do not matter.)

Q3. cos x + 7 (dy/dx)⁴ = 1: what is the degree of this differential equation?

• A. 3
• B. 1
• C. 4(correct answer)
• D. 2

Answer: C.The highest-order derivative is of order 1 and it appears raised to the power 4, so the degree is 4. (Powers of lower derivatives do not matter.)

Q4. x² (dy/dx)³ + x²y + 5 (d²y/dx²)² = eˣ: what is the degree of this differential equation?

• A. 4
• B. 2(correct answer)
• C. 1
• D. 3

Answer: B.The highest-order derivative is of order 2 and it appears raised to the power 2, so the degree is 2. (Powers of lower derivatives do not matter.)

Q5. 5 dy/dx + 3 d²y/dx² = sin x: what is the degree of this differential equation?

• A. 1(correct answer)
• B. 2
• C. 4
• D. 3

Answer: A.The highest-order derivative is of order 2 and it appears raised to the power 1, so the degree is 1. (Powers of lower derivatives do not matter.)

Q6. 3y + (dy/dx)⁴ = 0: what is the degree of this differential equation?

• A. 3
• B. 1
• C. 4(correct answer)
• D. 2

Answer: C.The highest-order derivative is of order 1 and it appears raised to the power 4, so the degree is 4. (Powers of lower derivatives do not matter.)

Q7. Is the degree defined for d²y/dx² + sin(dy/dx) = 0?

• A. Yes, the degree is 2 for this equation
• B. No, the equation is not polynomial in dy/dx(correct answer)
• C. Yes, the degree is 0 for this equation
• D. Yes, the degree is 1 for this equation

Answer: B.The derivative dy/dx appears inside sin, so the equation cannot be written as a polynomial in the derivatives; the degree is not defined (standard knowledge).

Q8. Is f(x, y) = sin(y/x) homogeneous? If so, of what degree?

• A. Homogeneous of degree 2
• B. Homogeneous of degree 0(correct answer)
• C. Not homogeneous
• D. Homogeneous of degree 1

Answer: B.Replacing (x, y) by (tx, ty) multiplies the expression by t to the power 0 (or leaves it unchanged for degree 0), so it is homogeneous of degree 0.

Q9. Is f(x, y) = (x² + y²)/xy homogeneous? If so, of what degree?

• A. Homogeneous of degree 1
• B. Homogeneous of degree 0(correct answer)
• C. Homogeneous of degree 2
• D. Not homogeneous

Answer: B.Replacing (x, y) by (tx, ty) multiplies the expression by t to the power 0 (or leaves it unchanged for degree 0), so it is homogeneous of degree 0.

Q10. Is f(x, y) = y/x + 1 homogeneous? If so, of what degree?

• A. Homogeneous of degree 1
• B. Homogeneous of degree 2
• C. Homogeneous of degree 0(correct answer)
• D. Not homogeneous

Answer: C.Replacing (x, y) by (tx, ty) multiplies the expression by t to the power 0 (or leaves it unchanged for degree 0), so it is homogeneous of degree 0.