SOLVING SECOND ORDER NON HOMOGENEOUS DIFFERENTIAL EQUATIONS

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MM - 455 Differential Equations

4.1 n^th-order Linear Equations

Homogeneous

An n^th-order linear differential equation is homogeneous if it can be written in the form:

The word homogeneous here does not mean the same as the homogeneous coefficients of chapter 2. Here it refers to the fact that the linear equation is set to 0. The general solution to this differential equation is y = c₁y₁( x ) + c₂y₂( x ) + ... + c_ny_n( x ), where each of the y_i( x ) are linearly independent. To be linearly independent means that none of the equations can be written as a linear combination of the others. For example, y₁( x ) = e^x and y₂( x ) = 5e^x are not linearly independent since the second is 5 times the first, whereas y₁( x ) = e^x and y₂( x ) = 5e^-x are linearly independent.

Wronskians

The usual method for determining linear independence is with a Wronskian. For two functions, f1( x ) and f2( x ), the Wronskian is defined to be: . If the determinant is 0 the functions are dependent, otherwise they are independent. For f1( x ) = e^x and f2( x ) = 5e^x, the Wronskian = 0. For the functions f1( x ) = e^x and f2( x ) = 5e^-x, the Wronskian = -10, showing that these are linearly independent. For three functions of x, the Wronskian is:

.

Therefore for an n^th-order linear homogeneous differential equation,we are looking for an n-parameter solution of linearly independent functions of x.

For example, using DSolve{ } to solve the second order differential equation x²y'' - 3xy' + 4y = 0, use the usual:

. Mathematica will return the proper two parameter solution of two linearly independent solutions.

Non-Homogeneous

An n^th-order linear differential equation is non-homogeneous if it can be written in the form:

The only difference is the function g( x ). The general solution to this differential equation is y = c₁y₁( x ) + c₂y₂( x ) + ... + c_ny_n( x ) + y_p, where y_p is a particular solution. The first part is identical to the homogeneous solution of above. The general method for solving non-homogeneous differential equations is to solve the homogeneous case first and then solve for the particular solution that depends on g( x ). The sum of the two is the general solution. Generally the solution is written as y( x ) = y_c( x ) + y_p( x ), where y_c( x ), the complementary solution, is the solution to the homogeneous differential equation and y_p( x ), the particular solution, is a solution based on g( x ).

 

Now, the question is will Mathematica be able to solve these non-homogeneous cases in y=the correct form? The answer is it can and will for all those cases that we normally do by hand and then some. Let's solve . The solution should be of the same form as above and a particular solution with no parameters ( constants ) added to it. The solution returned is exactly that:

In section 4.2 we will learn how to reduce the order of homogeneous linear differential equations if one solution is known. In section 4.3 we will solve all homogeneous linear differential equations with constant coefficients. In section 4.5 we will solve the non-homogeneous case. For all other sections of chapter 4, a CAS program should be utilized. As usual we will use Mathematica's DSolve[ ], but your test will be based on the techniques in 4.2, 4.3 and 4.5.

 

Things to know and do

• Find the Wronskian to determine linear independence of several functions of x.
• Use Mathematica to solve homogeneous and non-homogeneous differential equations.
• Determine the linear independence of y = 5, y = sin²( x ), y = cos²( x ) with a Wronskian.
• Solve the non-homogeneous differential equation x²y'' + xy' + y = x.