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.

 

[blue_raver22]

Hi there,

I understand your confusion about the different notations used in different books for the particular solution in Variation of Parameters. Let me try to explain it to you.

Firstly, let's understand what the particular solution represents. In the method of Variation of Parameters, we are trying to find a particular solution that satisfies the original non-homogeneous differential equation. This particular solution is added to the general solution of the associated homogeneous equation to get the complete solution.

Now, coming to the different notations for the particular solution, both the indefinite integral and the definite integral are correct. The indefinite integral with the integration constant equal to 0 is used when we are solving the problem in a general sense. This means that we are not given any specific initial conditions and we are trying to find a solution that satisfies the given differential equation for all values of x.

On the other hand, the definite integral with limits of integration is used when we have specific initial conditions given. In this case, we are solving the problem for a specific value of x (denoted by x_0) and we need to find the particular solution that satisfies the given differential equation for that particular value of x.

In summary, both notations are correct and it just depends on the context of the problem. If specific initial conditions are given, we use the definite integral with limits of integration. Otherwise, we use the indefinite integral with the integration constant equal to 0.

I hope this helps clarify your doubts. If you have any further questions, please let me know. Happy studying!