Question

Solve the following differential equation:

(D² + 2D + 3)y = 0

Answer 1

The auxiliary equation is m² + 2m + 3 = 0

Here a = 1, b = 2, c = 3

m = `(-"b" +- sqrt("b"^2 - 4"ac"))/(2"a")`

= `(-2 +- sqrt((2)^2 - 4(1)(3)))/(2(1))`

= `(- 2 +- sqrt(4 - 12))/2`

= `(-2 +- sqrt(-8))/2`

= `(-2 +- sqrt(4 xx (-2)))/2`

= `(-2 +- 2sqrt(2)"i")/2`

= `(2(1 +- sqrt(2)"i"))/2`

m = `-1 + sqrt(2)"i"`

Let `alpha = - 1, beta = sqrt(2)`

The complementary function is

e^ax (Acosßx + Bsinßx)

∴ C.F = `"e"^-x ["A" cos sqrt(2)x + "B"sin sqrt(2)x]`

∴ The general solution is

y =  `"e"^-x ("A" cos sqrt(2)x + "B"sin sqrt(2)x)`