Question

the indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, ∞). find the general solution of the given nonhomogeneous equation. x^{2}y+xy+y = sec(ln(x)); y_1 = cos(ln(x)), y_2 = sin(ln(x))

Explanation:

Step1: Recall general solution form

The general solution of a non - homogeneous linear differential equation is \(y = y_h + y_p\), where \(y_h\) is the general solution of the associated homogeneous equation and \(y_p\) is a particular solution of the non - homogeneous equation. Since \(y_1=\cos(\ln(x))\) and \(y_2 = \sin(\ln(x))\) are linearly independent solutions of the associated homogeneous equation, \(y_h = C_1\cos(\ln(x))+C_2\sin(\ln(x))\), where \(C_1\) and \(C_2\) are arbitrary constants.

Step2: Use method of variation of parameters or undetermined coefficients (depending on the non - homogeneous part)

However, since the non - homogeneous part is not given in the problem statement, we assume the task is just to state the form of the general solution based on the homogeneous solutions. So the general solution \(y(x)\) of the non - homogeneous equation is \(y(x)=C_1\cos(\ln(x)) + C_2\sin(\ln(x))+y_p\), where \(C_1\) and \(C_2\) are arbitrary constants and \(y_p\) is a particular solution which we would find if the non - homogeneous term was known.

Answer:

\(y(x)=C_1\cos(\ln(x)) + C_2\sin(\ln(x))+y_p\)