Please help to check that answer is correct: 2 d^2y/dx^2 + 4dy/dx = 3x^2 + 5x

[PA3040D]

Hi Experts
Could you please review the correctness of the following answer and the steps used to solve the problem? Although answer is correct, I have some doubts about in the middle section where I mentioned "Reason " why we can not use 'Cx^2+Dx+E'. for Yp.... I hope my question is clear. Thank you."

 

[mario99]

Your initial guess for the particular solution is:

[imath]y_p(x) = Cx^2 + Dx + E[/imath]

But then, you have to change it to:

[imath]y_p(x) = Cx^2 + Dx + Ex = Cx^2 + Fx[/imath] because there is a constant in the complementary solution [imath]\left[ \ y_c(x) = c_1e^{-2x} + c_2 \ \right][/imath].

The complementary solution, [imath]y_c(x)[/imath] and the particular solution, [imath]y_p(x)[/imath] are two independent solutions of [imath]y(x)[/imath], so they cannot have the same term (in this case a constant).

 

[PA3040D]

Your initial guess for the particular solution is:

[imath]y_p(x) = Cx^2 + Dx + E[/imath]

But then, you have to change it to:

[imath]y_p(x) = Cx^2 + Dx + Ex = Cx^2 + Fx[/imath] because there is a constant in the complementary solution [imath]\left[ \ y_c(x) = c_1e^{-2x} + c_2 \ \right][/imath].

The complementary solution, [imath]y_c(x)[/imath] and the particular solution, [imath]y_p(x)[/imath] are two independent solutions of [imath]y(x)[/imath], so they cannot have the same term (in this case a constant).

I think it should be as I got below two lines
Cx^3 + Dx^2 + Ex isn't it or I may be wrong?

 

[mario99]

Why do you want to include [imath]x^3[/imath] in the particular solution when the largest power in the right side of the differential equation is [imath]2[/imath], [imath]( \ x^2 \ )[/imath]?

You should do it like this:

[imath]y(x)_p = Dx^2 + Ex[/imath]

 

[mario99]

Why do you want to include [imath]x^3[/imath] in the particular solution when the largest power in the right side of the differential equation is [imath]2[/imath], [imath]( \ x^2 \ )[/imath]?

You should do it like this:

[imath]y(x)_p = Dx^2 + Ex[/imath]

I am sorry. I was wrong. You are correct.

Your initial guess was:

[imath]y(x)_p = Cx^2 + Dx + E[/imath]

Because there is a constant in the complementary solution, we modify the guess by:

[imath]y(x)_p = x(Cx^2 + Dx + E) = Cx^3 + Dx^2 + Ex[/imath]