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Question
Solve the following differential equation:
(4D2 + 4D – 3)y = e2x
Solution
The auxiliary equation is
4m2 + 4m – 3 = 0
4m2 + 6m – 2m – 3 = 0
2m(2m + 3) – 1(2m + 3) = 0
(2m + 3)(2m – 1) = 0
2m = – 3, 2m = 1
m = `-3/2, 1/2`
Roots are real and different
The complementary function is
Aem1x + Bem2x
C.F = `"Ae"^(1/2x) + "Be^((-3)/2 x)`
P.I = `1/(4"D"^2 + 4"D" - 3) "e"^(2x)`
= `"e"^(2x)/(4(2)^2 + 4(2) - 3)`
= `"e"^(2x)/(16 + 8 - 3)`
= `"e"^(2x)/21`
The general solution is y = C.F + P.I
y = `"Ae"^(1/2x) + "Be"^((-3)/2 x) + "e"^(2x)/21`
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