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प्रश्न
Solve the following differential equation:
(D2 + 2D + 3)y = 0
उत्तर
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
eax (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)`
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