Question

In Method of Variation of Parameters, \( A = -\int \frac{y_2 R dx}{W} \). If \( V = \sin x, R = \sec x, W = 1 \) then A= ____ .

• A. Sinx
• B. Cosx
• C. Log(Sinx)
• D. Log(Cosx)

Hint:

Variation of Parameters. Used to find a particular solution \(y_p = u_1 y_1 + u_2 y_2\) for non-homogeneous linear ODEs. The coefficients are found by integrals involving the complementary solutions \(y_1, y_2\), the non-homogeneous term R(x), and the Wronskian W. Remember standard integrals like \(\int \tan x dx = -\ln|\cos x| + C\).

Answer 1

The Correct Option is D

The formula for A in the method of variation of parameters is given as \( A = -\int \frac{y_2 R dx}{W} \)
We are given: \( y_2 = V = \sin x \) (Assuming V represents the second solution \(y_2\)) \( R = \sec x \) (This usually represents the non-homogeneous term divided by the coefficient of y'') \( W = 1 \) (The Wronskian) Substitute these into the integral for A: $$ A = -\int \frac{(\sin x)(\sec x)}{1} dx $$ Recall that \( \sec x = 1 / \cos x \)
$$ A = -\int \frac{\sin x}{\cos x} dx $$ $$ A = -\int \tan x \, dx $$ The integral of \(\tan x\) is \( -\ln|\cos x| \) or \( \ln|\sec x| \)
$$ A = - (-\ln|\cos x|) + C_1 = \ln|\cos x| + C_1 $$ Assuming the constant of integration is incorporated elsewhere or we need the principal value, \( A = \ln(\cos x) \), which is often written as Log(Cosx)
This matches option (4)