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Find the general solution of the following: y '''+3y''-4y = 0.?
Thank you in advance!!
1 Answer
- ?Lv 74 years agoFavorite Answer
y''' + 3y'' − 4y = 0
If r is a single root of characteristic equation, then e^(rx) is a solution to the homogeneous equation.
If r is a double root of characteristic equation, then e^(rx) and x e^(rx) are solutions to the homogeneous equation [This can be expanded for roots with multiplicity n: e^(rx), x e^(rx), x² e^(rx), ..., xⁿ⁻¹ e^(rx)]
Ir r is a complex root: α±βi, then e^(αx) sin(βx) and e^(αx) cos(βx) are solutions to the homogeneous equation.
r³ + 3r² − 4 = 0
(r − 1) (r² + 4r + r) = 0
(r − 1) (r + 2)² = 0
r = 1 (single root), r = −2 (double root)
y = C₁ e^x + C₂ e^(−2x) + C₃ x e^(−2x)