Linear Differential Equations: Classification, Solutions, and Extensions

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What is a linear differential equation?

An equation in which the unknown function and its derivatives appear linearly: they are not multiplied together or put inside nonlinear functions. Its coefficients may still depend nonlinearly on the independent variable.

How is the order of an equation defined?

It is the highest derivative appearing in the equation with a coefficient that is not identically zero.

What does homogeneous mean for $Ly=b(x)$?

The equation is homogeneous when the term independent of the unknown function and its derivatives is zero, so $b(x)=0$.

What is the general form of a non-homogeneous solution?

It is $y=y_p+y_h$, where $y_p$ is one particular solution and $y_h$ is any solution of the associated homogeneous equation.

What does the kernel of $L$ represent?

The kernel is the set of functions mapped to zero by $L$. For a differential operator, these are precisely the solutions of the homogeneous equation $Ly=0$.

Quiz questions

1. If a characteristic root $r$ has multiplicity $m$, which functions does it contribute to a basis?Answer: $e^{rx}, xe^{rx}, \ldots, x^{m-1}e^{rx}$
2. For real coefficients, what real-valued terms arise from a simple pair of characteristic roots $\alpha \pm i\beta$, where $\beta>0$?Answer: $e^{\alpha x}\cos(\beta x)$ and $e^{\alpha x}\sin(\beta x)$
3. For the standard equation $y'+p(x)y=q(x)$, which expression is a standard integrating factor?Answer: $e^{\int p(x)\,dx}$
4. What condition defines a holonomic (D-finite) function?Answer: It satisfies a homogeneous linear differential equation with polynomial coefficients.
5. For a regular second-order linear ODE, what do two independent initial values at one point determine?Answer: A unique solution satisfying those initial values.