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Differential Equations Practice Problems with Solutions - A Level Year 13

A Level Year 13 | Math | Differential Equations
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A first-order equation involves dy ÷ dx and separating variables rearranges dy ÷ dx = f(x)g(y) into dy ÷ g(y) = f(x) dx, so this Year 13 sheet starts from the first derivative. The key also records y = Ceᵏˣ as a standard solution form.

Key Takeaways

Section A: Multiple Choice Questions (1 Mark Each)

Choose the correct option for each question.
1. A first-order differential equation involves
(a) dy ÷ dx(b) d²y ÷ dx²(c) ∫ y dx(d) y²
2. To separate the variables in dy ÷ dx = f(x)g(y), you rearrange the equation to
(a) dy ÷ g(y) = f(x) dx(b) dy = f(x) dx(c) g(y) dy = f(x) + dx(d) nothing can be done
3. The general solution of a first-order differential equation contains
(a) one arbitrary constant(b) two arbitrary constants(c) no constants(d) three constants
4. A particular solution is found by using
(a) an initial condition(b) the general solution(c) another differential equation(d) the integral sign
5. The solution of dy ÷ dx = ky (exponential growth) is
(a) y = Ceᵏˣ(b) y = Ckx(c) y = kC(d) y = k eˣ
6. An equation of the form d²y ÷ dx² + a dy ÷ dx + by = 0 is
(a) second-order linear(b) first-order(c) quadratic(d) non-linear

Section B: Short Answer Type Questions (2 Marks Each)

Show all steps clearly.
7. Define a differential equation.
8. What is the order of the differential equation dy ÷ dx = x²?
9. Write the general solution of dy ÷ dx = k.
10. What is a particular solution?
11. How many arbitrary constants does the general solution of a second-order differential equation contain?
12. What is an initial condition?
13. Give one physical use of differential equations.
14. What method is used to solve dy ÷ dx = f(x)g(y)?

Section C: Numericals & Word Problems (3 Marks Each)

Apply the concepts to solve the problems. Show all working.
15. Solve dy ÷ dx = 5x⁴ by integrating.
16. Solve dy ÷ dx = 2x given that y = 3 when x = 0.
17. Solve dy ÷ dx = eˣ given that y = 0 when x = 0.
18. Solve dy ÷ dx = 3x² + 2x by integrating.
19. A population grows according to dy ÷ dt = 0.05y. Write the general solution.
20. Rewrite dy ÷ dx = 1 ÷ x² as a power of x and integrate.

Answer Key

1. a) dy ÷ dx
2. a) dy ÷ g(y) = f(x) dx
3. a) one arbitrary constant
4. a) an initial condition
5. a) y = Ceᵏˣ
6. a) second-order linear
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 first-order differential equation involve?

dy ÷ dx, option (a) in this worksheet (Q1).

How do you separate variables in dy ÷ dx = f(x)g(y)?

Rearrange to dy ÷ g(y) = f(x) dx, option (a) in this worksheet (Q2).

What form does the exponential solution take?

y = Ceᵏˣ, from the answer key on this page.

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