Question on Variation of Parameters

[yungman]

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

The associate homogeneous solution .$$y_c=c_1y_1 + c_2y_2$$.

The particular solution .$$y_p=u_1y_1 + c_2y_2$$.

$$u'_1 = -\frac{W_1}{W} = -\frac{y_2f(x)}{W}$$

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

$$u_1= -\int \frac{y_2f(x)}{W}dx$$

But other books gave:

$$u_1= -\int_{x_0}^x \frac{y_2f(s)}{W}ds$$

Where $$x_0$$ is any number in I.

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

Thanks

 

[LCKurtz]

If y_p = u₁y₁ + u₂y₂ is a particular solution you found, then compare this to:

Y_p = (u₁+A)y₁ + (u₂+B)y₂

=u₁y₁ + u₂y₂ + Ay₁+ By₂

When you plug the second expression into the equation the last two terms give 0 because they satisfy the homogeneous equation. So the constants don't give anything extra.

 

[yungman]

If y_p = u₁y₁ + u₂y₂ is a particular solution you found, then compare this to:

Y_p = (u₁+A)y₁ + (u₂+B)y₂

=u₁y₁ + u₂y₂ + Ay₁+ By₂

When you plug the second expression into the equation the last two terms give 0 because they satisfy the homogeneous equation. So the constants don't give anything extra.

Thanks for the respond.

So you mean even using definite integral, substituding in x0 only produce a constant as in your example of A and B. These will be absorbed into the associate homogeneous part ( into c1 and c2).

 

[matematikawan]

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

$$u_1= -\int \frac{y_2f(x)}{W}dx$$

But other books gave:

$$u_1= -\int_{x_0}^x \frac{y_2f(s)}{W}ds$$

Where $$x_0$$ is any number in I.

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

Thanks

Does it matter? We only want a particular solution.