# Differential Equations

## Higher Order Equations # 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 or tap 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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Concept
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Problems 1-4