find the general solution ..
Y=Asin4x+Bcos4x+x
To find the general solution of the given equation Y = Asin(4x) + Bcos(4x) + x, we need to solve for the constants A and B.
First, let's rewrite the equation using the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b):
Y = Asin(4x) + Bcos(4x) + x
= A(sin(4x)cos(x)) + B(cos(4x)cos(x)) + x
Now, let's use another trigonometric identity sin(2a) = 2sin(a)cos(a) and cos(2a) = cos^2(a) - sin^2(a):
Y = A(2sin(2x)cos(2x)) + B(cos(4x)cos(x)) + x
= A(2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)) + B(cos(2x)cos(2x) - sin(2x)sin(2x)) + x
= A(2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)) + B(cos^2(2x) - sin^2(2x)) + x
Expanding further:
Y = A(2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)) + Bcos^2(2x) - Bsin^2(2x) + x
Now, let's focus on the term 2sin(x)cos(x)cos(2x). We can use the double-angle identity sin(2x) = 2sin(x)cos(x) to simplify it:
Y = A(2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)) + Bcos^2(2x) - Bsin^2(2x) + x
= A(2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)) + Bcos^2(2x) - B(1 - cos^2(2x)) + x
= A(2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)) + Bcos^2(2x) - B + Bcos^2(2x) + x
= A(2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)) + 2Bcos^2(2x) - B + x
Now, we can rearrange the terms:
Y = A[2sin(x)cos(x)cos(2x) + 2cos^2(x)sin(2x)] + 2Bcos^2(2x) - B + x
This is the general solution for the given equation.
To find the general solution for the equation Y = Asin(4x) + Bcos(4x) + x, we can start by rearranging the equation.
1. Group the sine and cosine terms together:
Y = Asin(4x) + Bcos(4x) + x
2. Rewrite the sine and cosine terms using their equivalent trigonometric identities:
Y = A(2sin(2x)cos(2x)) + B(cos^2(2x) - sin^2(2x)) + x
3. Simplify the equation:
Y = 2A*sin(2x)*cos(2x) + B*cos^2(2x) - B*sin^2(2x) + x
4. Use the double-angle formulas to further simplify:
Y = A*sin(4x) + B*(1 - 2*sin^2(2x)) + x
5. Distribute B:
Y = A*sin(4x) + B - 2B*sin^2(2x) + x
6. Rearrange the terms:
Y = A*sin(4x) - 2B*sin^2(2x) + x + B
Now, this is the general solution for the equation Y = Asin(4x) + Bcos(4x) + x, where A, B are arbitrary constants.