1 Point In This Problem You Will Solve The Non Homogeneous Differential Equation Y 9y Sec 3x On The Interval N 1 (65.99 KiB) Viewed 21 times
1 Point In This Problem You Will Solve The Non Homogeneous Differential Equation Y 9y Sec 3x On The Interval N 2 (56.65 KiB) Viewed 21 times
(1 point) In this problem you will solve the non- homogeneous differential equation y" +9y=sec² (3x) on the interval −ñ/6 < x < π/6. (1) Let C₁ and C₂ be arbitrary constants. The general solution of the related homogeneous differential equation y" +9y = 0 is the function Yh (x) = C₁ y₁ (x) + C₂ Y2(x) = C₁ +C₂ sin(3x) cos(3x) (2) The particular solution y(x) to the differential equation y" +9y = sec² (3x) is of the form yp(x) = y₁ (x) u₁(x) + y₂ (x) u₂(x) where u₁(x) = -(1/9)sec(3x) and u₂(x) = (1/9)In(tan(3x)+
P where u₁(x) = -(1/9)sec(3x) (1/9)In(tan(3x)+ JL\/" (3) It follows that u₁(x) = -1/(9cos(3x)) and u₂(x) = u₂(x) = (1/9)In(tan(3x)+sec(3x)) thus y(x) = (1/9) (sin(3x)*In(tan(3x)+sec(3x)-1)) and ; (4) Therefore, on the interval (-л/6, л/6), the most general solution of the non-homogeneous differential equation y" +9y = sec² (3x) is y = C₁ cos(3x) +C₂ sin(3x) (1/9) (sin(3x)*In(tan(3x)+sec(3x +
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