Order and Linearity

Precise definition

The order of a differential equation is the highest derivative of the unknown function that appears. An $n$th-order ODE is linear if it can be arranged as $a_n(x)y^{(n)}+\cdots+a_1(x)y'+a_0(x)y=g(x)$, where coefficients and forcing depend only on the independent variable and the unknown function and derivatives appear to first power without products.

Notation and mathematical language

A linear equation is homogeneous when $g(x)=0$ and nonhomogeneous when $g\ne0$. This differs from the first-order scaling term 'homogeneous equation.' An equation is autonomous when the independent variable does not appear explicitly. Degree is not generally defined when derivatives occur inside non-polynomial expressions.

Conceptual picture

Classification predicts available structure. Linearity permits superposition for the homogeneous problem; order predicts the dimension of a solution space under regular coefficients and the typical number of initial values. Autonomy enables phase-line or phase-plane analysis.

Conditions and key results

First simplify without dividing by an expression that may be zero, because division can change domain. Coefficients in a linear ODE may depend on $x$ but not on $y$. Terms like $yy'$, $\sin y$, $(y')^2$, or a coefficient $y$ multiplying $y''$ make the equation nonlinear.

A reliable strategy

1. Identify the dependent and independent variables and expand hidden derivatives if necessary.
2. Find the highest derivative actually present after legitimate simplification.
3. Test whether $y$ and its derivatives occur linearly with coefficients depending only on the independent variable.
4. Then record homogeneous/nonhomogeneous, autonomous/non-autonomous, coefficient type, and any singular points.

Fully worked example

Interpretation and application

Classification is a decision tool, not a solution. A nonlinear first-order equation can be easier than a high-order linear one, and variable coefficients may block elementary formulas. Still, the labels prevent applying superposition or characteristic roots where they are invalid.

Common mistakes

Verification and reasonableness

• Substitute $c_1y_1+c_2y_2$ into a claimed homogeneous linear operator and check superposition.
• Inspect coefficients to ensure none contains the dependent variable.
• Verify the order after expanding total derivatives and noting any cancellation is valid on the domain.

Practice

1. Classify $y''+y^2=0$.
2. Classify $y'+x y=0$.
3. Why is $y y'+1=0$ nonlinear?

Further deduction

For a regular $n$th-order homogeneous linear equation on an interval, the solution set is an $n$-dimensional vector space. The nonhomogeneous solutions do not form a vector space: adding two particular solutions doubles the forcing. Instead they form an affine set $y_p+\mathcal N(L)$, one particular solution plus any homogeneous solution. This distinction explains the complementary-plus-particular method.

Solving for the highest derivative gives a normal form only where its coefficient is nonzero. If that coefficient vanishes, the apparent order can change and an initial-value theorem based on the normalized equation cannot simply be extended across the singular point.

Related topics

• Introduction to Differential Equations
• Second-Order Linear Differential Equations
• Autonomous Equations