Differential Equations

Second Order Equations

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Second Order Linear Homogeneous Differential Equations with Variable Coefficients

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 \({\frac{{{y_1}\left( x \right)}}{{{y_2}\left( x \right)}}} \equiv \text{const,}\) 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, there is no general method for finding a particular solution. 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 or tap a problem to see the 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 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.

Example 2.

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

Solution.

The Wronskian of the system of two functions is calculated by the formula:

\[{W_{{y_1},{y_2}}}\left( x \right) = \left| {\begin{array}{*{20}{c}} {{y_1}\left( x \right)}&{{y_2}\left( x \right)}\\ {{y'_1}\left( x \right)}&{{y'_2}\left( x \right)} \end{array}} \right|.\]

Substituting the given functions and their derivatives, we obtain

\[{W_{{y_1},{y_2}}}\left( x \right) = \left| {\begin{array}{*{20}{c}} {\cos x}&{\sin x}\\ { - \sin x}&{\cos x} \end{array}} \right| = {\cos ^2}x + {\sin ^2}x = 1.\]

It follows from here, that functions \(\sin x\) and \(\cos x\) are linearly independent.

See more problems on Page 2.

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