Basic Concepts

An nth-order linear differential equation has the form $$${b}_{{n}}{\left({x}\right)}{{y}}^{{{\left({n}\right)}}}+{b}_{{{n}-{1}}}{\left({x}\right)}{{y}}^{{{\left({n}-{1}\right)}}}+\ldots+{b}_{{2}}{\left({x}\right)}{y}''+{b}_{{1}}{\left({x}\right)}{y}'+{b}_{{0}}{\left({x}\right)}{y}={g{{\left({x}\right)}}}$$$, where $$${g{{\left({x}\right)}}}$$$ and all coefficients $$${b}_{{j}}{\left({x}\right)},j={\overline{{{0}..{n}}}}$$$ depend solely on the variable $$${x}$$$. In other words, they do not depend on $$${y}$$$ or on any derivative of $$${y}$$$.

If $$${g{{\left({x}\right)}}}\equiv{0}$$$, the differential equation is called homogeneous; otherwise, it is called non-homogeneous.

For example, $$${{x}}^{{3}}{y}'''+\frac{{1}}{{x}}{y}''+{2}{y}'={0}$$$ is homogeneous, and $$${y}''+{x}{y}'={\cos{{\left({x}\right)}}}$$$ is non-homogeneous.

A linear differential equation has constant coefficients, if all the coefficients $$${b}_{{j}}{\left({x}\right)}$$$ are constants; if one or more of these coefficients are not constant, the differential equation has variable coefficients.

For example, $$${{y}}^{{{\left({4}\right)}}}+{{x}}^{{2}}{y}'+{x}{y}={6}$$$ has variable coefficients, and $$${3}{y}''+{y}'={{e}}^{{x}}$$$ has constant coefficients.

Theorem. Consider an initial value problem given by a linear differential equation and $$${n}$$$ initial conditions $$${y}{\left({x}_{{0}}\right)}={y}_{{0}}$$$, $$${y}'{\left({x}_{{0}}\right)}={y}_{{0}}'$$$, $$${y}''{\left({x}_{{0}}\right)}={y}_{{0}}''$$$, ..., $$${{y}}^{{{\left({n}-{1}\right)}}}{\left({x}_{{0}}\right)}={{y}_{{0}}^{{{n}-{1}}}}$$$. If $$${g{{\left({x}\right)}}}$$$ and $$${b}_{{j}}{\left({x}\right)},j={\overline{{{0}..{n}}}}$$$ are continuous in some interval $$${I}$$$ containing $$${x}_{{0}}$$$ and if $$${b}_{{n}}{\left({x}\right)}\ne{0}$$$ in $$${I}$$$, the given differential equation together with the initial conditions has a unique (only one) solution defined throughout $$${I}$$$.

When the conditions on $$${b}_{{n}}{\left({x}\right)}$$$ in the theorem hold, we can divide the differential equation by $$${b}_{{n}}{\left({x}\right)}$$$ to obtain $$${{y}}^{{{\left({n}\right)}}}+{a}_{{{n}-{1}}}{\left({x}\right)}{{y}}^{{{\left({n}-{1}\right)}}}+\ldots+{a}_{{2}}{\left({x}\right)}{y}''+{a}_{{1}}{\left({x}\right)}{y}'+{a}_{{0}}{\left({x}\right)}{y}=\phi{\left({x}\right)}$$$, where $$${a}_{{j}}{\left({x}\right)}=\frac{{{a}_{{j}}{\left({x}\right)}}}{{{b}_{{n}}{\left({x}\right)}}},j={\overline{{{0}..{n}-{1}}}}$$$ and $$$\phi{\left({x}\right)}=\frac{{{g{{\left({x}\right)}}}}}{{\phi{\left({x}\right)}}}$$$.

Now, let's define the differential operator $$${L}{\left({y}\right)}$$$ by $$${L}{\left({y}\right)}={{y}}^{{{\left({n}\right)}}}+{a}_{{{n}-{1}}}{\left({x}\right)}{{y}}^{{{\left({n}-{1}\right)}}}+\ldots+{a}_{{2}}{\left({x}\right)}{y}''+{a}_{{1}}{\left({x}\right)}{y}'+{a}_{{0}}{\left({x}\right)}{y}$$$; then, the differential equation can be rewritten as $$${L}{\left({y}\right)}=\phi{\left({x}\right)}$$$, and, in particular, the linear homogeneous differential equation can be expressed as $$${L}{\left({y}\right)}={0}$$$.

The differential operator has two properties:

If $$${L}{\left({y}\right)}={0}$$$, we have that $$${L}{\left({c}{y}\right)}={0}$$$ for any constant $$${c}$$$.

If $$${L}{\left({y}_{{1}}\right)}={0}$$$ and $$${L}{\left({y}_{{2}}\right)}={0}$$$, we have that $$${L}{\left({y}_{{1}}+{y}_{{2}}\right)}={0}$$$.

These two properties can be combined into one property: if $$${L}{\left({y}_{{1}}\right)}={0}$$$, $$${L}{\left({y}_{{2}}\right)}={0}$$$, ..., $$${L}{\left({y}_{{n}}\right)}={0}$$$, we have that $$${L}{\left({c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}+\ldots+{c}_{{n}}{y}_{{n}}\right)}={0}$$$ for any constants $$${c}_{{i}},i={\overline{{{1}..{n}}}}$$$.

What does it give us? It gives us the following fact: if we have $$${n}$$$ solutions $$${y}_{{1}}$$$, $$${y}_{{2}}$$$, ..., $$${y}_{{n}}$$$ that satisfy the given homogeneous differential equation, their linear combination will also satisfy this homogeneous differential equation. The only question is what form $$${y}_{{1}}$$$, $$${y}_{{2}}$$$, ..., $$${y}_{{n}}$$$ should have for their linear combination to provide the general solution of the homogeneous differential equation.

Example. $$${y}_{{1}}={\cos{{\left({t}\right)}}}$$$ and $$${y}_{{2}}={\sin{{\left({t}\right)}}}$$$ are solutions of the differential equation $$${y}''+{y}={0}$$$. So, $$${y}_{{{g}}}={c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}={c}_{{1}}{\cos{{\left({t}\right)}}}+{c}_{{2}}{\sin{{\left({t}\right)}}}$$$ is also the solution for any constants $$${c}_{{1}}$$$ and $$${c}_{{2}}$$$.