Integral

If (D + 16)y = cos 4x

Here the auxiliary equation is m + 16 = 0
m = 4

Complementary function = (A cos 4x + B sin 4x) &
Particular Integral (P.I.)

But

P.I.= . sin 4x
=

Solution y = Complementary function Particular Integral
y = A cos 4x + B sin 4x + sin 4 x

Explanation:


On differentiating w.r.t. y, we get

Step 1: Finding the slope of the given line.
Rewriting the equation in slope-intercept form: So, the slope of this line is .
Step 2: Finding the slope of the normal.
The normal is perpendicular to this line, so its slope is the negative reciprocal:
Step 3: Finding the equation of the normal line.
Let the equation of the normal line be: To check which point does not lie on this line, substitute each given point and verify.
Checking point : If this equation does not hold for any valid , then does not lie on the normal. After solving for different valid points on the normal, we find that (10, -4) does not satisfy the equation, so it is not a valid point. Thus, the correct answer is (C) .