A-level Further Maths Year 13 · Topic 4
Differential equations
Differential equations model anything that changes over time.
We will solve first-order equations with an integrating factor and second-order equations with constant coefficients.
Then we model simple harmonic motion, damping and coupled systems such as predators and prey.
- 5 subtopics
- 64 questions
- Free: 4.1 notes and 10 questions
In this topic
- 4.1First-order differential equations: the integrating factorFree below · 14 questions
- 4.2Second-order differential equations with constant coefficientsOpen in Full Marks · 12 questions
- 4.3Simple harmonic motion and damped oscillationsOpen in Full Marks · 13 questions
- 4.4Coupled first-order systems: predator–prey and other modelsOpen in Full Marks · 13 questions
- 4.5Modelling with differential equationsOpen in Full Marks · 12 questions
4.1 · Free notes
First-order differential equations: the integrating factor
Key points
- A first-order linear differential equation has the form dy/dx + P(x)y = Q(x).
- Multiply through by the integrating factor I(x) = e^(∫P dx). The left side becomes d/dx(I y), so I y = ∫I Q dx, then divide by I.
- Example: dy/dx + 2y = e^(−x): I = e^(2x); d/dx(e^(2x)y) = eˣ; e^(2x)y = eˣ + c; y = e^(−x) + ce^(−2x).
- The general solution includes the constant c; a particular solution uses a boundary condition to find c.
- If the equation is not in standard form (for example x dy/dx + y = x²), divide by the coefficient of dy/dx first.
In more detail
- Worked example: dy/dx − y tan x = cos x: I = e^(∫−tan x dx) = cos x; d/dx(y cos x) = cos²x; y cos x = x/2 + (sin 2x)/4 + c.
- Modelling: Newton's law of cooling dT/dt = −k(T − T₀) is first order and can be solved by separating variables or with an integrating factor.
- Exam tip: always check the coefficient of dy/dx is 1 before finding the integrating factor.
Practice
First-order differential equations: the integrating factor: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is the integrating factor for dy/dx + 3y = x?
- e^(x)
- 3x
- e^(x²)
- e^(3x)
Show answer
D: e^(3x). e^(∫3 dx).
2. What is the integrating factor for dy/dx + (1/x)y = x (x > 0)?
- e^x
- x
- ln x
- 1/x
Show answer
B: x. e^(ln x) = x.
3. After multiplying by I(x), the left side becomes …
- d/dx(I y)
- I dy/dx
- dy/dx
- I y
Show answer
A: d/dx(I y). That is why it works.
4. Solve dy/dx + 2y = 0.
- y = ce^(2x)
- y = 2x + c
- y = ce^(−2x)
- y = c
Show answer
C: y = ce^(−2x). Separable or I = e^(2x).
5. Put x dy/dx + 2y = x³ into standard form.
- dy/dx + 2y = x³
- dy/dx + 2xy = x⁴
- x dy/dx = x³ − 2y
- dy/dx + (2/x)y = x²
Show answer
D: dy/dx + (2/x)y = x². Divide by x.
6. For dy/dx + (2/x)y = x², what is the integrating factor?
- e^(2x)
- x²
- 2 ln x
- 1/x²
Show answer
B: x². e^(2 ln x) = x².
7. Solve dy/dx + (2/x)y = x².
- y = x³/5 + c/x²
- y = x³ + c
- y = x⁵/5 + c
- y = cx²
Show answer
A: y = x³/5 + c/x². d/dx(x²y) = x⁴, so x²y = x⁵/5 + c.
8. What distinguishes a particular solution from the general solution?
- It has no x
- It is always zero
- A boundary condition fixes the constant
- It uses a different method
Show answer
C: A boundary condition fixes the constant. c is found from the given condition.
9. dy/dx + y = 1 with y = 0 when x = 0. What is y?
- e^(−x)
- 1 + e^(−x)
- x
- 1 − e^(−x)
Show answer
D: 1 − e^(−x). y = 1 + ce^(−x), c = −1.
10. Which equation is first-order linear?
- (dy/dx)² + y = 0
- dy/dx + x²y = sin x
- dy/dx = y²
- d²y/dx² + y = 0
Show answer
B: dy/dx + x²y = sin x. y and dy/dx appear to the first power only.
Keep going in Full Marks
Full Marks has 14 questions on First-order differential equations: the integrating factor (4 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Differential equations, plus games, timed exams and past papers.