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Basic Concepts

An nth-order linear differential equation has the form bn(x)y(n)+bn−1(x)y(n−1)+…+b2(x)y′′+b1(x)y′+b0(x)y=g(x){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(x){g{{\left({x}\right)}}} and all coefficients bj(x),j=0..n‾{b}_{{j}}{\left({x}\right)},j={\overline{{{0}..{n}}}} depend solely on the variable x{x}. In other words, they do not depend on y{y} or on any derivative of y{y}.

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

For example, x3y′′′+1xy′′+2y′=0{{x}}^{{3}}{y}'''+\frac{{1}}{{x}}{y}''+{2}{y}'={0} is homogeneous, and y′′+xy′=cos⁡(x){y}''+{x}{y}'={\cos{{\left({x}\right)}}} is non-homogeneous.

A linear differential equation has constant coefficients, if all the coefficients bj(x){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(4)+x2y′+xy=6{{y}}^{{{\left({4}\right)}}}+{{x}}^{{2}}{y}'+{x}{y}={6} has variable coefficients, and 3y′′+y′=ex{3}{y}''+{y}'={{e}}^{{x}} has constant coefficients.

Theorem. Consider an initial value problem given by a linear differential equation and n{n} initial conditions y(x0)=y0{y}{\left({x}_{{0}}\right)}={y}_{{0}}, y′(x0)=y0′{y}'{\left({x}_{{0}}\right)}={y}_{{0}}', y′′(x0)=y0′′{y}''{\left({x}_{{0}}\right)}={y}_{{0}}'', ..., y(n−1)(x0)=y0n−1{{y}}^{{{\left({n}-{1}\right)}}}{\left({x}_{{0}}\right)}={{y}_{{0}}^{{{n}-{1}}}}. If g(x){g{{\left({x}\right)}}} and bj(x),j=0..n‾{b}_{{j}}{\left({x}\right)},j={\overline{{{0}..{n}}}} are continuous in some interval I{I} containing x0{x}_{{0}} and if bn(x)≠0{b}_{{n}}{\left({x}\right)}\ne{0} in I{I}, the given differential equation together with the initial conditions has a unique (only one) solution defined throughout I{I}.

When the conditions on bn(x){b}_{{n}}{\left({x}\right)} in the theorem hold, we can divide the differential equation by bn(x){b}_{{n}}{\left({x}\right)} to obtain y(n)+an−1(x)y(n−1)+…+a2(x)y′′+a1(x)y′+a0(x)y=ϕ(x){{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 aj(x)=aj(x)bn(x),j=0..n−1‾{a}_{{j}}{\left({x}\right)}=\frac{{{a}_{{j}}{\left({x}\right)}}}{{{b}_{{n}}{\left({x}\right)}}},j={\overline{{{0}..{n}-{1}}}} and ϕ(x)=g(x)ϕ(x)\phi{\left({x}\right)}=\frac{{{g{{\left({x}\right)}}}}}{{\phi{\left({x}\right)}}}.

Now, let's define the differential operator L(y){L}{\left({y}\right)} by L(y)=y(n)+an−1(x)y(n−1)+…+a2(x)y′′+a1(x)y′+a0(x)y{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(y)=ϕ(x){L}{\left({y}\right)}=\phi{\left({x}\right)}, and, in particular, the linear homogeneous differential equation can be expressed as L(y)=0{L}{\left({y}\right)}={0}.

The differential operator has two properties:

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

If L(y1)=0{L}{\left({y}_{{1}}\right)}={0} and L(y2)=0{L}{\left({y}_{{2}}\right)}={0}, we have that L(y1+y2)=0{L}{\left({y}_{{1}}+{y}_{{2}}\right)}={0}.

These two properties can be combined into one property: if L(y1)=0{L}{\left({y}_{{1}}\right)}={0}, L(y2)=0{L}{\left({y}_{{2}}\right)}={0}, ..., L(yn)=0{L}{\left({y}_{{n}}\right)}={0}, we have that L(c1y1+c2y2+…+cnyn)=0{L}{\left({c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}+\ldots+{c}_{{n}}{y}_{{n}}\right)}={0} for any constants ci,i=1..n‾{c}_{{i}},i={\overline{{{1}..{n}}}}.

What does it give us? It gives us the following fact: if we have n{n} solutions y1{y}_{{1}}, y2{y}_{{2}}, ..., yn{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 y1{y}_{{1}}, y2{y}_{{2}}, ..., yn{y}_{{n}} should have for their linear combination to provide the general solution of the homogeneous differential equation.

Example. y1=cos⁡(t){y}_{{1}}={\cos{{\left({t}\right)}}} and y2=sin⁡(t){y}_{{2}}={\sin{{\left({t}\right)}}} are solutions of the differential equation y′′+y=0{y}''+{y}={0}. So, yg=c1y1+c2y2=c1cos⁡(t)+c2sin⁡(t){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 c1{c}_{{1}} and c2{c}_{{2}}.