Higher Order Linear Homogeneous Differential Equations with Variable Coefficients


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Higher Order Linear Homogeneous Differential Equations with Variable Coefficients

The linear homogeneous equation of the \(n\)th order has the form

\[{y^{\left( n \right)}} + {a_1}\left( x \right){y^{\left( {n - 1} \right)}} + \cdots + {a_{n - 1}}\left( x \right)y' + {a_n}\left( x \right)y = 0,\]

where the coefficients \({a_1}\left( x \right),\) \({a_2}\left( x \right), \ldots ,\) \({a_n}\left( x \right)\) are continuous functions on some interval \(\left[ {a,b} \right].\)

The left side of the equation can be written in abbreviated form using the linear differential operator \(L:\)

\[Ly\left( x \right) = 0,\]

where \(L\) denotes the set of operations of differentiation, multiplication by the coefficients \({a_i}\left( x \right),\) and addition.

The operator \(L\) is linear, and therefore has the following properties:

  1. \(L\left[ {{y_1}\left( x \right) + {y_2}\left( x \right)} \right] =\) \( L\left[ {{y_1}\left( x \right)} \right] + L\left[ {{y_2}\left( x \right)} \right],\)
  2. \(L\left[ {Cy\left( x \right)} \right] =\) \( CL\left[ {y\left( x \right)} \right],\)

where \({{y_1}\left( x \right)},\) \({{y_2}\left( x \right)}\) are arbitrary, \(n - 1\) times differentiable functions, \(C\) is any number.

It follows from the properties of the operator \(L\) that if the functions \({y_1},{y_2}, \ldots ,{y_n}\) are solutions of the homogeneous differential equation of the \(n\)th order, then the function of the form

\[y\left( x \right) = {C_1}{y_1} + {C_2}{y_2} + \cdots + {C_n}{y_n},\]

where \({C_1},{C_2}, \ldots ,{C_n}\) are arbitrary constants, will also satisfy this equation.

The last expression is the general solution of homogeneous differential equation if the functions \({y_1},{y_2}, \ldots ,{y_n}\) form a fundamental system of solutions.

Fundamental System of Solutions

The set of \(n\) linearly independent particular solutions \({y_1},{y_2}, \ldots ,{y_n}\) is called a fundamental system of the homogeneous linear differential equation of the \(n\)th order.

The functions \({y_1},{y_2}, \ldots ,{y_n}\) are linearly independent on the interval \(\left[ {a,b} \right]\) if the identity

\[{\alpha _1}{y_1} + {\alpha _2}{y_2} + \cdots + {\alpha _n}{y_n} \equiv 0\]

holds only provided

\[{\alpha _1} = {\alpha _2} = \cdots = {\alpha _n} = 0,\]

where the numbers \({\alpha _1},{\alpha _2}, \ldots ,{\alpha _n}\) are not simultaneously \(0.\)

To test functions for linear independence it is convenient to use the Wronskian:

\[W\left( x \right) = {W_{{y_1},{y_2}, \ldots ,{y_n}}}\left( x \right) = \left| {\begin{array}{*{20}{c}} {{y_1}}&{{y_2}}& \cdots &{{y_n}}\\ {{y'_1}}&{{y'_2}}& \cdots &{{y'_n}}\\ \cdots & \cdots & \cdots & \cdots \\ {y_1^{\left( {n - 1} \right)}}&{y_2^{\left( {n - 1} \right)}}& \cdots &{y_n^{\left( {n - 1} \right)}} \end{array}} \right|.\]

Let the functions \({y_1},{y_2}, \ldots ,{y_n}\) be \(n - 1\) times differentiable on the interval \(\left[ {a,b} \right].\) Then if these functions are linearly dependent on the interval \(\left[ {a,b} \right],\) then the following identity holds:

\[W\left( x \right) \equiv 0.\]

Accordingly, if these functions are linearly independent on \(\left[ {a,b} \right],\) we have the formula

\[W\left( x \right) \ne 0.\]

The fundamental system of solutions uniquely defines a linear homogeneous differential equation. In particular, the fundamental system \({y_1},{y_2},{y_3}\) defines a third-order equation, which is expressed through determinant as follows:

\[\left| {\begin{array}{*{20}{c}} {{y_1}}&{{y_2}}&{{y_3}}&y\\ {{y'_1}}&{{y'_2}}&{{y'_3}}&y'\\ {{y^{\prime\prime}_1}}&{{y^{\prime\prime}_2}}&{{y^{\prime\prime}_3}}&y^{\prime\prime}\\ {{y^{\prime\prime\prime}_1}}&{{y^{\prime\prime\prime}_2}}&{{y^{\prime\prime\prime}_3}}&y^{\prime\prime\prime} \end{array}} \right| = 0.\]

The expression for the differential equation of the \(n\)th order can be written similarly:

\[\left| {\begin{array}{*{20}{c}} {{y_1}}&{{y_2}}& \cdots &{{y_n}}&y\\ {{y'_1}}&{{y'_2}}& \cdots &{{y'_n}}&y'\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ {y_1^{\left( n \right)}}&{y_2^{\left( n \right)}}& \cdots &{y_n^{\left( n \right)}}&{{y^{\left( n \right)}}} \end{array}} \right| = 0.\]

Liouville's Formula

Suppose that the functions \({y_1},{y_2}, \ldots ,{y_n}\) form a fundamental system of solutions for a differential equations of \(n\)th order. Suppose that the point \({x_0}\) belongs to the interval \(\left[ {a,b} \right].\) Then the Wronskian is determined by Liouville's formula:

\[W\left( x \right) = W\left( {{x_0}} \right){e^{ - \int\limits_{{x_0}}^x {{a_1}\left( t \right)dt} }},\]

where \({a_1}\) is the coefficient of the derivative \({y^{\left( {n - 1} \right)}}\) in the differential equation. Here we assume that the coefficient \({a_0}\left( x \right)\) of \({y^{\left( n \right)}}\) in the differential equation is equal to \(1.\) Otherwise, Liouville's formula takes the form:

\[W\left( x \right) = W\left( {{x_0}} \right){e^{ - \int\limits_{{x_0}}^x {\frac{{{a_1}\left( t \right)}}{{{a_0}\left( t \right)}}dt} }},\;\; {a_0}\left( t \right) \ne 0,\;\; t \in \left[ {a,b} \right].\]

Reduction of Order of a Homogeneous Linear Equation

The order of a linear homogeneous equation

\[Ly\left( x \right) = {y^{\left( n \right)}} + {a_1}\left( x \right){y^{\left( {n - 1} \right)}} + \cdots + {a_{n - 1}}\left( x \right)y' + {a_n}\left( x \right)y = 0\]

can be reduced by one by the substitution \(y' = yz.\) Unfortunately, usually such a substitution does not simplify the solution, because the new equation in the variable \(z\) becomes nonlinear.

If a particular solution \({y_1}\) is known, then the order of the differential equation can be reduced (while maintaining its linearity) by replacing

\[y = {y_1}z,\;\;z' = u.\]

In general, if we know \(k\) linearly independent particular solutions, the order of the equation can be reduced by \(k\) units.