Solve The Following Differential Equation Y 4y Sin3x
Problems with Differential Equations? 3
Can you help me with this problem and explain the steps to me? Thank you very much.
Solve the following subtraction equations:
y + 4y = sin
First, consider the genetic equation:
y + 4y = 0
Of course this is a simple harmonic escalator, the complementary solutions are sin 2x and cos 2x.
Now we have to find a special solution. But using sin 3x is a difficult format, so let's change it:
sin x 3 x = sin x (sin 2 x) = sin x (1/2) (1 cos 2x)
= (1/2) sin x (1/2) sin x cos 2x
= (1/2) sin x (1/4) [sin 3x + sin (x)]
= (1/2) sin x (1/4) sin 3x + (1/4) sin x
= (3/4) sin x (1/4) sin 3x.
This is a method we know how to handle and it does not collide with complementary solutions, so we test the solution one way or another.
y_p = A sin x + B cos x + C sin 3x + D cos 3x
y_p = A cos x B sin x + 3C cos 3x 3D sin 3x
y_p = A sin x B cos x 9C sin 3x 9D cos 3x
And then y_p + 4y_p
= 3A sin x + 3B cos x 5C sin 3x 5D cos 3x
So we need 3A = 3/4, 3B = 0.5C = 1/4, 5D = 0
And so the specific solution is y_p = (1/4) sin x + (1/20) sin 3x
And collect everything we have.
y (x) = (1/4) sin x + (1/20) sin 3x + C_1 sin 2x + C_2 cos 2x.
