Hello, I have the DE: \(\displaystyle y'' + 2y'+y=2x \cdot sin(x)\) and I proceed to solve using the method of undetermined coefficients and some work I got the following equation and I am not sure how to solve for the coefficients:

\(\displaystyle [ \, \alpha \, (2mx+2b+1) \, + \, 2\beta \,] \, cos(x) + [\, \beta \, (-2mx-b-1) \,+ \,2\alpha \,] \, sin(x)=2x \cdot sin(x)\)

where \(\displaystyle \alpha \, , \beta \, , m \, ,b \, \) are real coefficients.

The solution manual solved this problem using variation of parameters.

Thank you for your time,

-Sakon

\(\displaystyle [ \, \alpha \, (2mx+2b+1) \, + \, 2\beta \,] \, cos(x) + [\, \beta \, (-2mx-b-1) \,+ \,2\alpha \,] \, sin(x)=2x \cdot sin(x)\)

where \(\displaystyle \alpha \, , \beta \, , m \, ,b \, \) are real coefficients.

The solution manual solved this problem using variation of parameters.

Thank you for your time,

-Sakon

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