Find the general solution to the following linear differential equations with constant coefficients.

Question

a. y'''−3y''−y'+3y = 0

b. 2y'''−y''−10y'−7y = 0

c. y'''−9y''+27y'−27y = 0

d. y'''−6y''+2y'+36y = 0

Answer 1

a) Characteristic equation: r³ -3r² - r + 3 = 0 (r - 3)(r² - 1) = 0; r = 3, r = ±1

So, y = C₁e^3x + C₂e^x + C₃e^-x

b) CharacterisC₁e^3x + C₂e^x + C₃e^-xtic equation: 2r³ - r² - 10r - 7 = 0; (r + 1)(2r² -3r - 7) = 0; r = - 1, or r = (3 ± √65)/4

So, y = C₁e^-x + C₂e^(3+√65)/4x + C₃e^{(3 - √65)/4x}

c) Characteristic equation : r³ - 9r² + 27r - 27 = 0; (r - 3)(r² - 6r + 9) = 0; (r - 6)³ = 0; r = 3 multiplicity 3;

So y = (ax² + bx + c)e^3x

d) Characteristic equation r³ - 6r² + 2r + 36 = 0; (r + 2)(r² - 8r + 18) = 0; r = - 2 or r = 4 ±i√2

y = C₁e^-2x + e^4x(C₂cos(x√2) + C₃sin(x√2))