Differential Equations

Higher Order Equations

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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} }},\;\;}\kern-0.3pt
    {{a_0}\left( t \right) \ne 0,\;\;}\kern-0.3pt{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.

  • Solved Problems

    Click a problem to see the solution.

    Example 1

    Show that the functions \(x,\) \(\sin x,\) \(\cos x\) are linearly independent.

    Example 2

    Show that the functions \(x,{x^2},{x^3},{x^4}\) form a linearly independent system.

    Example 3

    Make a differential equation, which is determined by the fundamental system of functions \(1,{x^2},{e^x}.\)

    Example 4

    Find the general solution of the equation \(\left( {2x – 3} \right)y^{\prime\prime\prime} \) \(-\; \left( {6x – 7} \right)y^{\prime\prime} \) \(+\; 4xy’ – 4y = 0,\) if the particular solutions \({y_1} = {e^x},\) \({y_2} = {e^{2x}}\) are known.
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    Problems 1-4