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 \({\large\frac{{{y_1}\left( x \right)}}{{{y_2}\left( x \right)}}\normalsize} \equiv \text{const,}\) 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 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 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