Classifying ODEs: order, linearity & solution type
Classification decides the method, so it comes first. The order is the highest derivative present. An ODE is linear when the unknown function and its derivatives appear only to the first power and are never multiplied together — the coefficients may be any functions of the independent variable. Homogeneous means the forcing term is zero.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Find the orderThe highest derivative appearing. y″ + y = 0 is second order regardless of the powers elsewhere.
- Test linearityLook for y², yy′, sin y or (y′)³. Any of these makes it non-linear. Terms like x²y′ are fine — the coefficient may depend on x freely.
- Check homogeneityMove everything involving y to the left. If the right side is 0 the equation is homogeneous.
- Distinguish general from particular solutionsThe general solution of an nth-order ODE has n arbitrary constants; initial conditions pin them down.
Worked example
Classify y″ + 3x y′ − y = e^x and (y′)² + y = 0.
- First: highest derivative is y″, so order 2.
- y, y′ and y″ all appear linearly; the coefficient 3x depends only on x, which is allowed. So it is linear, and inhomogeneous since the right side is e^x.
- Second: highest derivative is y′, so order 1.
- (y′)² is a squared derivative, so it is non-linear.
Answer. The first is second order, linear, inhomogeneous. The second is first order and non-linear.
Where marks get dropped
These are the specific errors that cost credit on classifying odes: order, linearity & solution type questions — QED's rubric penalises each of them separately.
- Calling an equation non-linear because a coefficient involves x. Linearity constrains how y and its derivatives appear, not the coefficients.
- Confusing "homogeneous" in the linear sense (zero right side) with "homogeneous" as a first-order substitution type (functions of y/x). Both terms are standard and they mean different things.
- Counting the number of arbitrary constants wrongly. An nth-order equation needs exactly n.
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Classifying ODEs: order, linearity & solution type — frequently asked questions
Why does linearity matter so much?
Because linear equations obey superposition: solutions add. That gives the entire general-solution-plus-particular-solution structure, which non-linear equations lack.
What is the difference between an ODE and a PDE?
An ODE has one independent variable, so only ordinary derivatives. A PDE has several and uses partial derivatives.
How many initial conditions do I need?
One per order: a second-order equation needs y(x₀) and y′(x₀) to determine a unique solution.
The rest of Differential Equations
Solving and modelling with ODEs, from separable to systems. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Classifying ODEs: order, linearity & solution type
- 2Separable equations
- 3First-order linear equations & the integrating factor
- 4Initial value problems & particular solutions
- 5Substitutions: homogeneous & Bernoulli
- 6Exact equations & integrating factors
- 7Second-order linear homogeneous equations
- 8Undetermined coefficients
- 9Variation of parameters
- 10Modelling: growth, decay, cooling & mixing
- 11Oscillations, damping & resonance
- 12Slope fields, equilibria & qualitative behaviour
- 13Systems of linear ODEs via eigenvalues
- 14Laplace transforms for initial value problems
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