# Differential Equations

Nth Order Equations# Higher Order Linear Homogeneous Differential Equations with Variable Coefficients

Theory

Problems 1-4

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

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 \(L:\)

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:

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

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

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

holds only provided

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:

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

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:

{{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:

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

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

## Solved Problems

Click on problem description to see 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.

Theory

Problems 1-4

### Related Pages

- Higher Order Linear Homogeneous Differential Equations with Constant Coefficients
- Higher Order Linear Nonhomogeneous Differential Equations with Constant Coefficients
- Higher Order Linear Nonhomogeneous Differential Equations with Variable Coefficients
- Second Order Linear Homogeneous Differential Equations with Constant Coefficients
- Second Order Linear Nonhomogeneous Differential Equations with Constant Coefficients
- Second Order Linear Homogeneous Differential Equations with Variable Coefficients
- Second Order Linear Nonhomogeneous Differential Equations with Variable Coefficients