For a second-order linear ODE 3y''-6y'+6y=e^xsec x, the method of variation of parameters requires finding the particular solution y_p = u_1 y_1 + u_2 y_2 where u_1' and u_2' satisfy the system involving the:

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SSC Applied Mathematics QUESTION #1300

Question 1

For a second-order linear ODE \(3y''-6y'+6y=e^x\sec x\), the method of variation of parameters requires finding the particular solution \(y_p = u_1 y_1 + u_2 y_2\) where \(u_1'\) and \(u_2'\) satisfy the system involving the:

Wronskian of \(y_1\) and \(y_2\)✔️
Coefficient of \(y''\) only
Initial conditions
Characteristic equation roots only

Correct Answer Explanation

In variation of parameters, \(u_1'\) and \(u_2'\) are found by solving: \(u_1'y_1+u_2'y_2=0\) and \(u_1'y_1'+u_2'y_2'=g(x)\), where \(g(x)\) is the right-hand side (after dividing by the leading coefficient). The solutions use the Wronskian \(W=y_1y_2'-y_2y_1'\) in the denominators.

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