MTH2032 Chap.1 First-Order ODEs: Separable, Linear, Exact and Homogeneous
First-Order ODEs: Separable, Linear, Exact and Homogeneous
Week 1 introduces the four first-order families that come with a guaranteed solution procedure: separable, linear, exact and homogeneous type. The skill being trained is recognition. Once an equation is placed in the right family, the method is routine, and the marks go to stating the family, checking the conditions you rely on, and not losing solutions when you divide.
This chapter sets out the order in which to test an equation and works each method with original examples. Each method is shown with the check that confirms the answer, so you can verify your own work in a quiz.
What this chapter covers
- 01
Recognising separable equations
- 02
Standard form of a linear equation
- 03
Variation of parameter and formula (2.2)
- 04
The cross-derivative test for exactness
- 05
Finding a potential function
- 06
The substitution y = vx for ratio equations
Worked example · free
A linear equation with a constant forcing term
- 1Rewrite as y′ − 2y = 4: first-order linear with p(x) = −2 and q(x) = 4.
- 1The homogeneous solution uses γ = e^(2x); set y_p = vγ so that v′e^(2x) = 4 and v = −2e^(−2x).
- 1Then y_p = −2 and the general solution is y = Ce^(2x) − 2.
- 1The condition y(0) = 1 gives C − 2 = 1, so C = 3 and y = 3e^(2x) − 2.
Key terms
- Separable Equation
- A first-order equation that can be written g(y) dy = f(x) dx and integrated on each side.
- Integrating Structure
- The function γ = exp(−∫p dx) that solves the homogeneous linear equation and anchors variation of parameter.
- Potential Function
- A function u(x, y) whose partial derivatives equal M and N, so that solutions satisfy u = C.
- Homogeneous-Type Equation
- A first-order equation y′ = g(y/x) whose right-hand side depends only on the ratio of the variables.
- Implicit Solution
- A relation such as u(x, y) = C that defines the solution curves without isolating y as a formula in x.
First-Order ODEs: Separable, Linear, Exact and Homogeneous FAQ
How do I tell which first-order method to use?
Ask the questions in a fixed order: can the right-hand side split into a function of x times a function of y; can the equation be rearranged as y′ + py = q with coefficients in x; does M_y equal N_x; does everything depend on y/x. The first yes chooses the method, although some equations fit two families and either route works.
Why do markers care about dividing by y?
Separating variables usually divides by a function of y. That step assumes the function is not zero, so constant solutions where it vanishes can disappear. The problem-set solutions for this week award marks for stating the assumption and for checking afterwards whether a constant such as y = 0 also solves the equation.
Is a homogeneous-type equation the same as a homogeneous linear one?
No. The lecture notes separate the two ideas. A homogeneous-type equation has the ratio form y′ = g(y/x) and is solved with the substitution y = vx. A homogeneous linear equation is a linear equation whose right-hand side is zero. Confusing them sends you down the wrong method from the very first line.
What should the first line of a first-order solution say?
Name the family and give the reason, for example that the equation is linear because it rearranges to y′ − 2y = 4 with coefficients depending only on x. That single sentence tells the marker which method follows, shows you checked the form, and protects you if a later algebra slip hides an otherwise correct approach.
Exam move
Make a one-page recognition chart for the four families with the signal, the first line you would write and the usual slip, then test it on mixed equations without labels. For each equation, decide the family within thirty seconds before doing any algebra; this is the habit the timed applied-class quizzes reward.
Practise variation of parameter until you can write formula (2.2) from the idea rather than from memory: the homogeneous solution with its constant turned into a function. For exact equations, always integrate M in x and then match N, and keep the unknown function g(y) visible until it is found. For ratio equations, substitute back for v at the end and leave the answer in x and y.
Finally, after every separable solution, ask what happened to the values you divided by. Keeping a short list of such lost constant solutions from your own practice is a quick way to stop repeating the mistake. When a question asks for an explicit solution, rearrange the implicit relation only where it is valid and state the interval on which the formula holds.
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