Please solve as many as you can! Will thumbs up work! In the following problems

Question

Please solve as many as you can! Will thumbs up work!

In the following problems a complete set of roots (repeated by multiplicity) is given for the characteristic equation of an nth-order linear homogeneous differential equation in y(x) with real numbers as coefficients. Determine the general solution of the differential equation and write it in its real form. 1. 0, plusorminus 19i 2. 0, 0, 2 plusorminus 9i 3. -3 plusorminus i, -3 plusorminus i, -3 plusorminus i, -3 plusorminus i 4. plusorminus 6i, plusorminus 6i, plusorminus 6i

Explanation / Answer

1. 0, 19i, -19i

The solution is y(x) = a + be19ix + c-19ix

Using Euler's formula,

y(x) = a + b (cos 19x + i sin 19x) + c (cos 19x - i sin 19x)

=> y(x) = a + (b+c) cos 19x + i (b-c) sin 19x

2. 0,0,2+9i,2-9i

The roots are repeated.

The general solution is y(x) = a + dx + be(2+9i)x + ce(2-9i)x

=> y(x) = a + dx + be2xe9ix + ce2xe-9ix

Using Euler's formula

y(x) = a + dx + be2x (cos 9x + i sin 9x) + ce2x (cos 9x - i sin 9x)

=> y(x) = a + dx + (b+c)e2x cos 9x + i (b-c)e2x sin 9x

3. (This question is ambigious. Does this imply -3+i and -3-i four times each or four times together?)

4. (This has the same problem as 3. Assuming 6i three times and -6i three times the solution is below)

y(x) = ae6ix + bxe6ix + cx2e6ix + de-6ix + exe-6ix + fx2e-6ix

Using Euler's formula,

y(x) = a (cos 6x + i sin 6x) + bx (cos 6x + i sin 6x) + cx2 (cos 6x + i sin 6x) +  d (cos 6x - i sin 6x) + ex (cos 6x - i sin 6x) + fx2 (cos 6x - i sin 6x)

y(x) = (a+bx+cx2+d+ex+fx2) cos 6x + i (a+bx+cx2-d-ex-fx2) sin 6x

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