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In this theorem we identify that for a specified differential equation a set of fundamental solutions will exist.
Consider the differential equation
y′′ + p (t ) y′ + q (t ) y = 0
Here p(t) and q(t) are continuous functions on any interval I. select t0 to be any point in the interval I. Let y1(t) be a solution to the differential equation which satisfies the initial conditions.
y(t0) = 1
y′ (t0) = 0
Let y2(t) be a solution to the differential equation which satisfies the initial conditions.
y (t0) = 0
y′ (t0) = 1
So y1(t) and y2(t) form a fundamental set of solutions for the differential equation.
This is easy enough to illustrate that these two solutions form a fundamental set of solutions. Just calculate the Wronskian.
Thus, fundamental sets of solutions will exist; we can solve the two IVP's specified in the theorem.
Two trains leave two different cities 1,029 miles apart and head directly toward every other on parallel tracks. If one train is traveling at 45 miles per hour and the other at 53
01010011 01100101 01101101 01110000 01100101 01110010 00100000 01000110 01101001 00100001
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