374: Solving Homogeneous Linear Differential Equations with Constant Coefficients

Transcription

1 374: Solving Homogeneous Linear Differential Equations with Constant Coefficients Enter this Clear command to make sure all variables we may use are cleared out of memory. In[1]:= Clearx, y, c1, c2, c3, c4, c5, c6, Y Homogeneous Linear Differential Equations Consider the nth order linear homogeneous differential equation with constant coefficients of the form b 0 y n b n-1 y ' + b n y = 0. (1) The general solution to (1) is y = c 1 y 1 + c 2 y c n y n, where y 1, y 2,..., y n is a linearly independent set of solutions of (1). To find the solutions y 1, y 2,..., y n, we write (1) in the form f D y = 0 where f D = b 0 D n b n-1 D + b n and find the roots of the auxiliary equation f m = 0. To check that the n solutions y 1, y 2,..., y n that we find are linearly independent, we can compute the Wronskian of y 1, y 2,..., y n. Example 1: Solve y 6-11 y y y''' y'' - 65 y' = 0 Find the general solution. This differential equation can be written in differential operator form as D 6-11 D D D D 2-65 D y = 0. Thus, the auxiliary equation is m 6-11 m m m m 2-65 m = 0. To find the roots of the auxiliary equation, we can use Mathematica's Factor or Solve commands. Factor will attempt to factor a polynomial into a product of binomial and irreducible quadratic terms. Solve will solve an equation for specified variables. Notice that in the Solve command, we include the equal sign (the syntax in Mathematica is = =). Factor left-hand side of the auxiliary equation: In[2]:= Out[2]= Factorm^6 11 m^5 92 m^4 220 m^3 203 m^2 65 m 1 m 3 m 65 8m m 2 Find the roots of the auxiliary equation and form the general solution: In[3]:= Out[3]= Solvem^6 11 m^5 92 m^4 220 m^3 203 m^2 65 m 0, m m 0, m 1, m 1, m 1, m 4 7, m 4 7 Thus, the roots of the auxiliary equation are: m = 0, 1, 1, 1, 4 ± 7 Â. The general solution to this differential equation is: y = c 1 + c 2 x + c 3 x x + c 4 x 2 x + c 5 4 x cos 7 x + c 6 4 x sin 7 x.

2 2 374SolvingHomogeneousLinearEquationsWithConstantCoefficientsFall2017.nb Check linear independence of the "different" solutions that make up the general solution via the Wronskian. Check that the six "different" functions found are linearly independent and that the given general solution is actually a solution of the differential equation! The following set of commands saves the six functions in memory as FS, sets up the n x n matrix with rows corresponding to the functions and the first n-1 derivatives of each function, and defines the Wronskian to be the determinant of this matrix. To evaluate the Wronskian function at x, enter W[x]. Save the set of solutions in memory: In[4]:= FS 1, E^x, x E^x, x^2 E^x, E^4 x Cos7 x, E^4 x Sin7 x; Construct the n x n matrix of functions and derivatives up to the (n-1)st derivative: In[5]:= TableTableDFSi, x, j, i, 1, LengthFS, j, 0, LengthFS 1 MatrixForm Out[5]//MatrixForm= 1 x x x x x 2 4x Cos7 x 0 x x x x 2 x x x x 2 4 4x Cos7 x 7 4x Sin7 x 7 4x Cos 0 x 2 x x x 2 x 4 x x x x x Cos7 x 56 4x Sin7 x 56 4x Cos 0 x 3 x x x 6 x 6 x x x x x Cos7 x 7 4x Sin7 x 7 4x Cos 0 x 4 x x x 12 x 8 x x x x x Cos7 x x Sin7 x x Cos 0 x 5 x x x 20 x 10 x x x x x Cos7 x x Sin7 x x Cos In[6]:= The Wronskian is the determinant of this n x n matrix: Wx_ : Simplify DetTableTableDFSi, x, j, i, 1, LengthFS, j, 0, LengthFS 1 Check that the Wronskian is defined. If it is non-zero on an interval, the functions are linearly independent on that interval! In[7]:= Out[7]= Wx x Check that the general solution really is a solution to the differential equation. Finally, we can check that our general solution is indeed a solution to the differential equation by defining the general solution as a function in Mathematica, substituting the general solution into the left-hand side of the differential equation, and simplifying to see if we get the right-hand side! Define our general solution as a function in Mathematica: In[8]:= In[9]:= Out[9]= yx_ : c1 1 c2 E^x c3 x E^x c4 x^2 E^x c5 E^4 x Cos7 x c6 E^4 x Sin7 x Check that we defined the function correctly: yx c1 c2 x c3 x x c4 x x 2 c5 4x Cos7 x c6 4x Sin7 x Substitute our solution into the left-hand side of the differential equation:

3 374SolvingHomogeneousLinearEquationsWithConstantCoefficientsFall2017.nb 3 In[10]:= Out[10]= In[11]:= Out[11]= 0 Dyx, x, 6 11 Dyx, x, 5 92 Dyx, x, y'''x 203 y''x 65 y'x c2 x 6c3 x 30 c4 x c3 x x 12 c4 x x c4 x x c5 4x Cos7 x 7336 c6 4x Cos7 x 7336 c5 4x Sin7 x c6 4x Sin7 x 92 c2 x 4c3 x 12 c4 x c3 x x 8c4 x x c4 x x c5 4x Cos7 x 3696 c6 4x Cos7 x 3696 c5 4x Sin7 x 2047 c6 4x Sin7 x 220 c2 x 3c3 x 6c4 x c3 x x 6c4 x x c4 x x c5 4x Cos7 x 7c6 4x Cos7 x 7c5 4x Sin7 x 524 c6 4x Sin7 x 203 c2 x 2c3 x 2c4 x c3 x x 4c4 x x c4 x x 2 33 c5 4x Cos7 x 56 c6 4x Cos7 x 56 c5 4x Sin7 x 33 c6 4x Sin7 x 65 c2 x c3 x c3 x x 2c4 x x c4 x x 2 4c5 4x Cos7 x 7c6 4x Cos7 x 7c5 4x Sin7 x 4c6 4x Sin7 x 11 c2 x 5c3 x 20 c4 x c3 x x 10 c4 x x c4 x x c5 4x Cos7 x c6 4x Cos7 x c5 4x Sin7 x c6 4x Sin7 x Simplify our calculation (% means the last output). Simplify Example 2: Solve the initial value problem (IVP): y 6-11 y y y''' y'' - 65 y' = 0; y0 = 1; y' 0 = 0; y'' 0 =-1; y''' 0 = 5; y 4 0 =-2; y 5 0 = 3 Since on any interval, this differential equation is normal, it follows from Theorem 4.2 (Existence and Uniqueness) that with x 0 = 0 and y 0 = 1, y 1 = 0, y 2 =-1, y 3 = 5, y 4 =-2, y 5 = 3, this IVP has a unique solution. We've already found the general solution to the homogeneous problem, so all we have to do is apply our initial conditions to the general solution we found above in Example 1. Check to make sure our function for the general solution y, which we found above, is defined correctly: In[12]:= Out[12]= yx c1 c2 x c3 x x c4 x x 2 c5 4x Cos7 x c6 4x Sin7 x Find the choices of coefficients c 1, c 2,..., c 6 for the solution to this IVP. Apply the initial conditions at x = 0 to get a system of six equations in the six unknown coefficients c 1, c 2,..., c 6. Here are two ways to enter the same initial data and solve the resulting system. The first command uses prime notation for all the derivatives and the second command uses the replacement operator (/.) for the higher ordered derivatives. In the second command, the fourth and fifth derivatives of y are computed first, after which x is replaced by 0. In[13]:= Solvey0 1, y'0 0, y''0 1, y'''0 5, y''''0 2, y'''''0 3, c1, c2, c3, c4, c5, c6 Out[13]= c ,c2 13,c ,c4,c5 6427,c6 6161

4 4 374SolvingHomogeneousLinearEquationsWithConstantCoefficientsFall2017.nb In[14]:= Solvey0 1, y'0 0, y''0 1, y'''0 5, Dyx, x, 4. x 0 2, Dyx, x, 5. x 0 3, c1, c2, c3, c4, c5, c6 Out[14]= c ,c2 13,c ,c4,c5 6427,c Define our solution and check that it really is the solution to our IVP. In[15]:= Yx_ : E^x 9163 x E^x 571 x^2 E^x 6427 E^4 x Cos7 x 6161 E^4 x Sin7 x Check to make sure Y is defined correctly. In[16]:= Yx Out[16]= x 9163 x x x Cos7 x x Sin7 x Check that Y[x] is the solution to this IVP: In[17]:= Out[17]= DYx, x, 6 11 DYx, x, 5 92 DYx, x, Y'''x 203 Y''x 65 Y'x x x x x x x x x x x Cos7 x x Cos7 x x Cos7 x x Sin7 x x Sin7 x x Sin7 x 5295 x 1767 x x x Cos7 x x Sin7 x x x x x x x x Cos7 x x Cos7 x x Sin7 x x Sin7 x In[18]:= Out[18]= 0 Simplify In[19]:= Out[19]= 1 Y0 In[20]:= Out[20]= 0 Y'0 In[21]:= Out[21]= Y''0 1

5 374SolvingHomogeneousLinearEquationsWithConstantCoefficientsFall2017.nb 5 In[22]:= Out[22]= 5 Y'''0 In[23]:= DYx, x, 4. x 0 Out[23]= 2 In[24]:= DYx, x, 5. x 0 Out[24]= 3 In[25]:= PlotYx, x, 1, Out[25]= Using DSolve to Solve Differential Equations We can use the command DSolve to solve differential equations! The next two commands show how to solve Examples 1 and 2 using this command. The Clear command resets all variables we've used. If you are getting errors, make sure you enter this command. If you still get errors, close this notebook, restart Mathematica, ignore the first two sections and start with this section! Using DSolve to solve the differential equation: y 6-11 y y y''' y'' - 65 y' = 0 In[26]:= Clearx, y, c1, c2, c3, c4, c5, c6, Y In[27]:= DSolveDyx, x, 6 11 Dyx, x, 5 92 Dyx, x, y'''x 203 y''x 65 y'x 0, yx, x Out[27]= yx C x 65 C3 1 x C4 2 2xx 2 C5 3x 7C14C2 Cos7 x 3x 4C17C2 Sin7 x Compare this solution to the general solution we got above. Are they the same?

6 6 374SolvingHomogeneousLinearEquationsWithConstantCoefficientsFall2017.nb In[28]:= ExpandC x 65 C3 1 x C4 2 2xx 2 C5 3x 7C14C2 Cos7 x 3x 4C17C2 Sin7 x Out[28]= x C3 x C4 x xc42 x C5 2 x xc5 x x 2 C5 C x C1 Cos7 x x C2 Cos7 x x C1 Sin7 x x C2 Sin7 x Using DSolve to solve the IVP : y 6-11 y y y''' y'' - 65 y' = 0; y0 = 1; y' 0 = 0; y'' 0 =-1; y''' 0 = 5; y 4 0 =-2; y 5 0 = 3 In[29]:= Out[29]= DSolveDyx, x, 6 11 Dyx, x, 5 92 Dyx, x, y'''x 203 y''x 65 y'x 0, y0 1, y'0 0, y''0 1, y'''0 5, y''''0 2, y'''''0 3, yx, x 1 yx x x x x x x Cos7 x x Sin7 x Compare this solution to the IVP solution we got above. Are they the same? In[30]:= 1 Expand x x x x x x Cos7 x x Sin7 x Out[30]= x 9163 x x x Cos7 x x Sin7 x Homework The commands used above can also be used to solve a non-homogeneous linear differential equation. Using commands such as the ones given above, solve the following differential equation: D 2-4 y = 2 x + 2.