A differential equation involves derivatives of a function, and a first-order equation contains the first derivative dy ÷ dx, so this IB sheet defines its subject before solving. The key records a general solution y = x² + C and an initial-condition answer.
Key Takeaways
A differential equation involves derivatives of a function (Q1).
A first-order equation contains dy ÷ dx (Q2).
The key records y = x² + C as a general solution.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.A differential equation is an equation that involves
(a) derivatives of a function(b) only constants(c) only integers(d) square roots
2.A first-order differential equation contains
(a) the first derivative dy ÷ dx(b) the second derivative d²y ÷ dx²(c) an integral(d) only the function y
3.The general solution of a first-order differential equation contains
(a) one arbitrary constant(b) no constants(c) two arbitrary constants(d) infinitely many constants
4.To find the particular solution, you need
(a) an initial condition(b) nothing else(c) another equation(d) a graph
5.To separate variables in dy ÷ dx = f(x)g(y), you rearrange it to
(a) dy ÷ g(y) = f(x) dx(b) dy × g(y) = f(x) dx(c) g(y) dy = f(x) + dx(d) dy ÷ dx = f(x) + g(y)
6.The solution of dy ÷ dx = 2x is
(a) y = x² + C(b) y = 2x + C(c) y = x + C(d) y = 2x² + C
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Define a differential equation.
8.What is the order of a differential equation?
9.How many arbitrary constants are in the general solution of dy ÷ dx = f(x)?
10.What is an initial condition?
11.What method is used to solve dy ÷ dx = f(x)g(y)?
12.Write the general solution of dy ÷ dx = 3x².
13.What is a particular solution?
14.Give one physical application of differential equations.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.Solve dy ÷ dx = 4x³ by integrating.
16.Solve dy ÷ dx = 2x with the initial condition y = 5 when x = 0.
17.Solve dy ÷ dx = 6x² + 1 by integrating.
18.Solve dy ÷ dx = 3 given y = 2 when x = 0.
19.Solve dy ÷ dx = eˣ by integrating.
20.Solve dy ÷ dx = 1 ÷ x (for x > 0) by integrating.
Answer Key
1. a) derivatives of a function
2. a) the first derivative dy ÷ dx
3. a) one arbitrary constant
4. a) an initial condition
5. a) dy ÷ g(y) = f(x) dx
6. a) y = x² + C
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 does a differential equation involve?
Derivatives of a function, option (a) in this worksheet (Q1).
What does a first-order differential equation contain?
The first derivative dy ÷ dx, option (a) in this worksheet (Q2).
What general solution appears in the key?
y = x² + C.
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