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Question
Solve the following differential equation:
(3D2 + D – 14)y – 13e2x
Solution
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) em1x + Bem2x
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)`
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