Question

Solve the following differential equation:

(3D² + D – 14)y – 13e^2x

Answer 1

The auxiliary equation is 3m² + m – 14 = 0

3m² – 6m + 7m – 14 = 0

3m(m – 2) + 7(m – 2) = 0

(m – 2)(3m + 7) = 0

m = 2, 3m = – 7

m = 2, `(-7)/3`

Roots are real and different

C.F = (Ax + B) e^m₁x + Be^m₂x 

C.F = `"Ae"^(2x) + "Be"^((-7)/3 x)`

P.I = `1/((3"D"^2 + "D" - 14)) 13"e"^(2x)`

Replace D by 2

3D² + D – 4 = 0

When D = 2

∴ P.I = `x  1/((3(2"D") +1)) 13"e"^(2x)`

= `x * 1/((6"D" + 1)) 13"e"^(2x)`

= `x * 1/((6(2) + 1)) 13"e"^(2x)`

= `x * 1/13 13"e"^(2x)`

P.I = `x"e"^(2x)`

The general solution is y = C.F + P.I

y = `"Ae"^(2x) + "BE"^((-7)/2x) + x"e"^(2x)`