Suppose r; and r2 arc two complex numbers. Using Euler\'s formula exp(alpha + I

Question

Suppose r; and r2 arc two complex numbers. Using Euler's formula exp(alpha + I beta) = exp(alpha)(cos(beta) + i sin(beta)) where alpha, beta are real, show that: er1 er2 = er1 +r25) Substitute the expression y*(t) = f cxp(rt) into the equation ay" + br/ + cy = 0 where r satisfies ar7 + br + c 0 anti y is a function of t. Under what conditions of a, b, c will y2 be a solution of the DE?6a) Suppose that y: and y2 are solutions of the second order DE y" + + y fl(0 = 0. Show that Cyi + C2y2 is also a solution where C] and Cj arc arbitrary >' constants. This fact is called the principle of superposition and the expression Cy + C2y* is called a linear combination of yi and y*.6b) The DE in the problem 6a was of a special type. Can you think of DE where the principle of superposition docs not hold? (In other words, can you think of DE where a linear combination of two solutions is not a solution?).

Explanation / Answer

e^r1 e^r2

= e^(a+ib) e^(c+id)

= e^a ( cosb + i sinb) e^c (cosd + i sind)

= e^(a+c) ( cosb cosd - sinb sind + i sinb cosd + i cosb sind)

= e^(a+c) (cos(b+d) + i sin(b+d)

= e^(a+c+ib+id)

= e^(r1+r2)

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