Enzymes

Chapter 7: Linear Equations with Constant Coefficients

Introduction

* Several methods for solving differential equations with constant coefficients are presented in this book. A classical technique is treated in this and the next chapter. Chapters 14 and 15 contain a development of the Laplace transform and its use in solving linear differential equations. Each method has its advantages and disadvantages. Each is theoretically sufficient: all are necessary for maximum efficiency.

I. The Auxiliary Equation: Distinct Roots

1. Solve the equation

d3ydx3-4d2ydx2+dydx+6y=0

First write the auxiliary equation

m3-4m2+m+6=0

Whose roots m = 1, 2, 3 may be obtained by synthetic division. Then the general solution is seen to be

y= c1c1e-x+c2e2x+c3e3x

2. Solve the equation

3D3+5D2-2D)y=0

The auxiliary equation is

3m3+5m2-2m=0

And its roots are m = 0, -2, 1/3. By using the fact that e0x=1, the desired solution may be written

Y=c1+c2e-2x+c3exp13x

3. Solve the equation

d2xdt2-4=0

With the conditions that when t=0, x=0 and dx/dt=3.

The auxiliary equation is

m2 4 D 0;

With roots m D 2; 2. Hence the general solution of the differential equation is

x D c1 e2tC c2e2t:

It remains to enforce the conditions at t D 0. Now

dxdtD 2c1e2t 2c2e2t:

Thus the condition that x D 0 when t D 0 requires that

0 D c1 C c2;

And the condition that dx=dt D 3 when t D 0 requires that

3 D 2c1 2c2;

From the simultaneous equations for c1 and c2 we conclude that c1 D 34 and c2 D 34 Therefore,

X D 34.e2t e2t/;

Which can also be put in the form

x D 32sinh .2t ;/

Practice Set

1. .D2C 2D 3/y D 0: 11. d3xdt3Cd2xdt2 2dxdt D 0:

2. .D2C2DyD0: 12. . d3xdt319dxdt C 30x D 0:

3. .D2C D 6/y D 0: 13. .9D3 7D C 2/y D 0:

4. .D2 5DC6yD 0: 14. .4D3 21D 10/y D 0:

5. . D3C 3D2 4DyD 0: 15. .D3 14D C 8/y D 0:

6. . D3 3D2 10DyD 0: 16. .D3 D2 4D 2/y D 0:

7. . D3C6D2 C 11 D C 6/y D 0: 17. .4D4 8D3 7D2 C 11D C 6/y D 0:

8. .D3 C 3D2 4D 12/y D 0:...