Differential Equations

2nd Order Equations

Second Order Linear Homogeneous Differential Equations with Variable Coefficients

Page 1
Problem 1
Page 2
Problems 2-7

A linear homogeneous second order equation with variable coefficients can be written as

\[{y^{\prime\prime} + {a_1}\left( x \right)y’ }+{ {a_2}\left( x \right)y }={ 0,}\]

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

Linear Independence of Functions. Wronskian

The functions \({y_1}\left( x \right),{y_2}\left( x \right), \ldots ,{y_n}\left( x \right)\) are called linearly dependent on the interval \(\left[ {a,b} \right],\) if there are constants \({\alpha _1},{\alpha _2}, \ldots ,{\alpha _n},\) not all zero, such that for all values of \(x\) from this interval, the identity

\[{{\alpha _1}{y_1}\left( x \right) + {\alpha _2}{y_2}\left( x \right) + \ldots }+{ {\alpha _n}{y_n}\left( x \right) }\equiv {0}\]

holds. If this identity is satisfied only when \({\alpha _1} = {\alpha _2} = \ldots\) \( = {\alpha _n} = 0,\) then these functions \({y_1}\left( x \right), {y_2}\left( x \right), \ldots ,\) \({y_n}\left( x \right)\) are called linearly independent on the interval \(\left[ {a,b} \right].\)

For the case of two functions, the linear independence criterion can be written in a simpler form: The functions \({y_1}\left( x \right),\) \({y_2}\left( x \right)\) are linearly independent on the interval \(\left[ {a,b} \right],\) if their quotient in this segment is not identically equal to a constant:

\[\frac{{{y_1}\left( x \right)}}{{{y_2}\left( x \right)}} \ne \text{const.}\]

Otherwise, when \({\large\frac{{{y_1}\left( x \right)}}{{{y_2}\left( x \right)}}\normalsize} \equiv \text{const,}\) these functions are linearly dependent.

Let \(n\) functions \({y_1}\left( x \right),\) \({y_2}\left( x \right), \ldots ,\) \({y_n}\left( x \right)\) have derivatives of \(\left( {n – 1} \right)\) order. The determinant

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

is called the Wronski determinant or Wronskian for this system of functions.

Wronskian Test.
If the system of functions \({y_1}\left( x \right),\) \({y_2}\left( x \right), \ldots ,\) \({y_n}\left( x \right)\) is linearly dependent on the interval \(\left[ {a,b} \right],\) then its Wronskian vanishes on this interval.

It follows from here that if the Wronskian is nonzero at least at one point in the interval \(\left[ {a,b} \right],\) then the functions \({y_1}\left( x \right),\) \({y_2}\left( x \right), \ldots ,\) \({y_n}\left( x \right)\) are linearly independent. This property of the Wronskian allows to determine whether the solutions of a homogeneous differential equation are linearly independent.

Fundamental System of Solutions

A set of two linearly independent particular solutions of a linear homogeneous second order differential equation forms its fundamental system of solutions.

If \({y_1}\left( x \right),{y_2}\left( x \right)\) is a fundamental system of solutions, then the general solution of the second order equation is represented as

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

where \({C_1}, {C_2}\) are arbitrary constants.

Note that for a given fundamental system of solutions \({y_1}\left( x \right),\) \({y_2}\left( x \right)\) we can construct the corresponding homogeneous differential equation. For the case of a second order equation, it is expressed in terms of the determinant:

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

Liouville’s Formula

Thus, as noted above, the general solution of a homogeneous second order differential equation is a linear combination of two linearly independent particular solutions \({y_1}\left( x \right),\) \({y_2}\left( x \right)\) of this equation.

Obviously, the particular solutions depend on the coefficients of the differential equation. The Liouville formula establishes a connection between the Wronskian \(W\left( x \right),\) constructed on the basis of particular solutions \({y_1}\left( x \right),\) \({y_2}\left( x \right),\) and the coefficient \({a_1}\left( x \right)\) in the differential equation.

Let \(W\left( x \right)\) be the Wronskian of the solutions \({y_1}\left( x \right),\) \( {y_2}\left( x \right)\) of a linear second order homogeneous differential equation

\[{y^{\prime\prime} + {a_1}\left( x \right)y’ }+{ {a_2}\left( x \right)y }={ 0,}\]

in which the functions \({a_1}\left( x \right)\) and \({a_2}\left( x \right)\) are continuous on the interval \(\left[ {a,b} \right].\) Let the point \({x_0}\) belong to the interval \(\left[ {a,b} \right].\) Then for all \(x \in \left[ {a,b} \right]\) the Liouville formula

\[{W\left( x \right) }={ W\left( {{x_0}} \right)\exp \left( { – \int\limits_{{x_0}}^x {{a_1}\left( t \right)dt} } \right)}\]

is valid.

Practical methods for solving second order homogeneous equations with variable coefficients

Unfortunately, the general method of finding a particular solution does not exist. Usually this is done by guessing.

If a particular solution \({y_1}\left( x \right) \ne 0\) of the homogeneous linear second order equation is known, the original equation can be converted to a linear first order equation using the substitution \(y = {y_1}\left( x \right)z\left( x \right)\) and the subsequent replacement \(z’\left( x \right) = u.\)

Another way to reduce the order is based on the Liouville formula. In this case, a particular solution \({y_1}\left( x \right)\) must also be known. The relevant examples are given below.

Solved Problems

Click on problem description to see solution.

 Example 1

Investigate whether the functions \({y_1}\left( x \right) = x + 2,\) \({y_2}\left( x \right) = 2x – 1\) are linearly independent.

 Example 2

Find the Wronskian of the system of functions \({y_1}\left( x \right) = \cos x,\) \({y_2}\left( x \right) = \sin x.\)

 Example 3

Write a homogeneous linear differential equation, if its fundamental system of solutions is known: \(x,{e^x}.\)

 Example 4

Find the general solution of the equation \({x^2}y^{\prime\prime} – 2xy’ \) \(+ 2y = 0,\) given the particular solution \({y_1} = x.\)

 Example 5

Find the general solution of the equation \(\left( {{x^2} + 1} \right)y^{\prime\prime} – 2y\) \( = 0.\)

 Example 6

Find the general solution of the equation \({x^2}y^{\prime\prime} – 4xy’ + 6y\) \(= 0\) using the Liouville formula. A particular solution of the equation is known and has the form: \({y_1} = {x^2}.\)

 Example 7

Find the general solution of the equation \({x^2}y^{\prime\prime} + xy’ – y\) \( = 0\) (for \(x \ne 0\) by the Liouville formula, if a particular solution is known: \({y_1} = x.\)

Example 1.

Investigate whether the functions \({y_1}\left( x \right) = x + 2,\) \({y_2}\left( x \right) = 2x – 1\) are linearly independent.

Solution.

We form the quotient of two functions:

\[
{\frac{{{y_1}\left( x \right)}}{{{y_2}\left( x \right)}} }
= {\frac{{x + 2}}{{2x – 1}} }
= {\frac{{x – \frac{1}{2} + \frac{5}{2}}}{{2x – 1}} }
= {\frac{{\frac{1}{2}\left( {2x – 1} \right) + \frac{5}{2}}}{{2x – 1}} }
= {\frac{1}{2} + \frac{5}{{2\left( {2x – 1} \right)}} }
= {\frac{1}{2} + \frac{5}{{4x – 2}}.}
\]

It is seen that this ratio is not equal to a constant, but depends on \(x.\) Hence, these functions are linearly independent.

Page 1
Problem 1
Page 2
Problems 2-7