Second-order non-homogeneous linear differential equation

[properman]

So I learned about these equations in my math class this spring, and it has been bothering me for some time that you are able to merely drop the offending term when there is overlap in the solutions. A quick example might help, seeing as I don't really know the true terms.

When finding the general solution for y''-2y'=x+2e^2 you end up with an overlap in yh and yp of Ce^x, and so when bringing it together you just ignore one of them.

Why is this a legitimate thing to do? Why does it not violate any rules?

I tried to figure it out myself using the above equation and multiplying the two equations through by x, resulting in xyh+xyp=(C1x+C2xe^2x)+(Ax^2+Bx+Cxe^x). Not to bore you with the particulars, but it resulted in x=x, which of course tells me nothing.

Can someone please enlighten me?

 

[psktam]

Hey there,

Let me see if I understand what you're saying.

So you say that for the differential equation you gave:

y" - 2y' = x + e²​

You found your homogeneous solution:

y_h = C₁ + C₂e^2x​

And your particular solution:

y_p = Ax + B + Ce^x​

I think you might have made a mistake in there somewhere, because I am not sure exactly where you got your Ce^x term in your particular solution. If you plug your particular solution back into your original differential equation, you should find out that:

Ce^x - 2Ce^x -2A = x + 2e²​

Which becomes:

-Ce^x-2A = x + 2e²​

This indicates that there is an error in your particular solution because there is no term on the left-hand side of the above equation that can deal with the x-term on the right hand side of the equation. Additionally, this also indicates that your coefficient for e^x is 0.

What you should have gotten for your particular solution for this equation is:

y_p = Ax² + Bx + C;​

But besides the point, for your original question, say that you have this equation:

y" - 2y' = e^2x + x​

So you find:

y_h = A + Be^2x and
y_p = Fx² + Cx + G + De^2x.
​

Your particular and homogeneous solutions overlap by that term e^x. What happens next is a little tricky. When we add the homogeneous and particular solutions together, we don't altogether "drop" any of the e^x solutions. What we do is we replace the term De^x with Dxe^x so that the particular solution now looks like y_p = x + Dxe^x. This seems even more random, but a helpful thing to notice is that the term De^x is already part of the homogeneous solution, which means the term will cancel to 0 once it is applied back into our differential equation:

(Fx² + Cx + G + De^2x)" - 2(Fx² + Cx + G + De^2x)' = x + e^2x
2F + 4De^2x - 4Fx - 2C - 4De^2x = x + e^2x
4Fx - 2C + 2F + 4De^2x - 4De^2x = x + e^2x
4Fx - 2C + 2F = x + e^2x
​

Now, we can solve for the x-term, but our De^x term has disappeared! That is part of the reason why we replace De^2x with Dxe^2x in our particular solution. To understand the actual reason why we do this, if you have taken linear algebra before, what you can do is think of the differential equation as a matrix equation, so you can represent this differential equation as Ax = b, where A is an mxn matrix, x is a 1xm vector, and b is a 1xn vector that is in the column space of A. If our differential equation is homogeneous, it turns out that b is equal to the zero vector, and so the vector x exists only in the nullspace of A, which is basically a collection of ALL the vectors that return the zero vector if multiplied with A. The nullspace is analogous to our homogeneous solution, which is a collection of ALL the solutions that return zero if applied to our differential equation. If our differential equation is non-homogeneous, however, then b is not equivalent to the zero vector, and so we have to find some vector x that is NOT in the nullspace in order to properly equate the expression Ax = b. As we found out in the above example, our homogeneous solution contained the term De^2x, which put it in the "nullspace" of the matrix associated with this differential equation. When we tried to guess a particular solution, we found that one of its terms was also De^2x, but as was just pointed out, this term is already located in the nullspace, and so to remove it from the nullspace, we multiply it by x in order to make it distinct from the homogeneous solution.

Of course, this explanation is MUCH more complicated and fascinating than what I could express here. Try to look up math books that will combine linear algebra with linear differential equations because they do a great job of explaining this in much better detail. This book could also help you, it's a math textbook all about differential equations: http://tutorial.math.lamar.edu/pdf/DE/DE_Complete.pdf

I hope this helped, and good luck.

 

[properman]

oh shoot, I messed up writing it. it should actually be y'' - 2y' = x + 2e^x

Sorry about that

 

[psktam]

Hm,

Then in that case, yeah, your particular solution is right. Note, though, that when you add the particular and homogeneous solutions together, you do NOT get rid of either of the exponential forms. One is e^x and the other is e^2x, so in this case, your full solution should take the form:

y = C₁ + C₂e^2x + Ax + B + Ce^x
​

and since C₁ + B is just another constant, we can simply rewrite the above expression as:

y = D + C₂e^2x + Ax + Ce^x
​

 

[blue_raver22]

I can provide a response to your question about second-order non-homogeneous linear differential equations. Dropping the offending term in the general solution is a legitimate approach in solving these types of equations because of the principle of superposition. This principle states that if two solutions, y1 and y2, satisfy the differential equation, then any linear combination of these solutions, such as y = ay1 + by2, will also satisfy the equation. In other words, the sum of two solutions is also a solution.

In your example, the general solution has two parts: the complementary solution (yh) and the particular solution (yp). The complementary solution is the solution to the associated homogeneous equation (y''-2y'=0), while the particular solution is a specific solution to the non-homogeneous equation (y''-2y'=x+2e^2). When there is an overlap between the two solutions, it means that the complementary solution is also a solution to the non-homogeneous equation. In this case, we can simply drop the particular solution and use only the complementary solution as the general solution.

This approach does not violate any rules because it still satisfies the differential equation and initial conditions. In fact, it simplifies the general solution and makes it easier to work with. Your attempt to multiply the two equations through by x is not an appropriate method for solving this type of equation and may lead to incorrect results. I would recommend studying the principle of superposition and practicing more examples to gain a better understanding of this concept.