Question

Give the general solution of a differential equation if the roots of its characteristic equation are as follows:

1. m1=4 m2=2
2. m1=3i m2=−3i
3. m1=−6 m2=−6 m3=−6
4. m1=1−8i m2=1+8i
5. m1=2i m2=−2i m3=2i m4=−2i

Answer 1

The general solutions of the differential equations are found by using the characteristic roots to form exponentials, sine, and cosine functions, with arbitrary constants determined by initial or boundary conditions.

Step-by-step explanation:

The general solutions for each case based on the roots of the characteristic equation are as follows:

• Case 1: m1 = 4, m2 = 2. The solution is y(x) = C1e4x + C2e2x, where C1 and C2 are arbitrary constants.
• Case 2: m1 = 3i, m2 = -3i. The solution is y(x) = C1cos(3x) + C2sin(3x).
• Case 3: m1 = m2 = m3 = -6. The solution is y(x) = (C1 + C2x + C3x2)e-6x.
• Case 4: m1 = 1 - 8i, m2 = 1 + 8i. The solution is y(x) = ex(C1cos(8x) + C2sin(8x)).
• Case 5: m1 = m2 = 2i, m3 = m4 = -2i. The solution is y(x) = C1cos(2x) + C2sin(2x) + C3x cos(2x) + C4x sin(2x), where the x terms are included due to the repeated roots.

In all the cases, 'C1', 'C2', 'C3', and 'C4' represent arbitrary constants that would be determined by initial conditions or boundary conditions.