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I have a question on the integration part of the Variation of Parameters. Given .[tex]y''+P(x)y'+Q(x)y=f(x)[/tex]

The associate homogeneous solution .[tex] y_c=c_1y_1 + c_2y_2[/tex].

The particular solution .[tex] y_p=u_1y_1 + c_2y_2[/tex].

[tex]u'_1 = -\frac{W_1}{W} = -\frac{y_2f(x)}{W} [/tex]

This is where I have question. Some books use indefinite integral with the integration constant equal 0.

[tex]u_1= -\int \frac{y_2f(x)}{W}dx[/tex]

But other books gave:

[tex]u_1= -\int_{x_0}^x \frac{y_2f(s)}{W}ds[/tex]

Where [tex]x_0[/tex] is any number in I.

None of the books explain this. Can anyone explain to me about this?

Thanks

The associate homogeneous solution .[tex] y_c=c_1y_1 + c_2y_2[/tex].

The particular solution .[tex] y_p=u_1y_1 + c_2y_2[/tex].

[tex]u'_1 = -\frac{W_1}{W} = -\frac{y_2f(x)}{W} [/tex]

This is where I have question. Some books use indefinite integral with the integration constant equal 0.

[tex]u_1= -\int \frac{y_2f(x)}{W}dx[/tex]

But other books gave:

[tex]u_1= -\int_{x_0}^x \frac{y_2f(s)}{W}ds[/tex]

Where [tex]x_0[/tex] is any number in I.

None of the books explain this. Can anyone explain to me about this?

Thanks

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