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# Differential Equations

2nd Order Equations

# Second Order Linear Homogeneous Differential Equations with Variable Coefficients

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Problem 1
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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.

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Problem 1
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Problems 2-7