In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.
The solutions for the given linear system are and .
Elimination Procedure for 2 × 2 Systems
To find a general solution for the system
Where and are polynomials in
Let us rewrite the system in operator form,
Multiply D on both sides of equation (3) then subtract equation (3) and (4) together one gets,
Since the auxiliary equation to the corresponding homogeneous equation is . The roots are r=1 and r=-2 .
Then, the homogeneous solution of u is;
Let us take the undetermined coefficients and assume that,
Now find the derivate the equation (7).
Substitute the derivation in equation (5).
Now, equalize the like terms.
So, … (8)
Use equations (6) and (8) to get,
Now, take equation (3).
Thus, the solutions for the given linear system are and .
Arms Race. A simplified mathematical model for an arms race between two countries whose expenditures for defense are expressed by the variables x(t) and y(t) is given by the linear system
Where a and b are constants that measure the trust (or distrust) each country has for the other. Determine whether there is going to be disarmament (x and y approach 0 as t increases), a stabilized arms race (x and y approach a constant as ), or a runaway arms race (x and y approach as ).
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