If cos²x is an integrating factor of the differential equation dy/dx + Py = Q, then P can be
The integrating factor is a function used to solve linear differential equations. It helps convert a non-exact equation into an exact one. For first-order linear differential equations, the integrating factor simplifies the process of finding the solution by making the equation easier to integrate and solve.
Class 12 Maths Chapter 9 on Differential Equations focuses on understanding the relationship between a function and its derivatives. It includes methods for solving first-order and higher-order differential equations. Real-world applications such as motion and growth are explored. This chapter is important for the CBSE Exam 2024-25 and helps in solving practical problems.
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Given the differential equation:
dy/dx + P y = Q
The integrating factor μ(x) is defined by:
μ(x) = exp(∫ P dx)
We are told that:
μ(x) = cos²x
Taking the natural logarithm of both sides:
ln μ(x) = ln(cos²x) = 2 ln|cos x|
Differentiate with respect to x:
d/dx [ln μ(x)] = d/dx [2 ln|cos x|] = 2 · (−tan x) = −2 tan x
But we also have:
d/dx [ln μ(x)] = P
Thus, we find:
P = −2 tan x
Therefore, the correct answer is:
−2 tan x
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