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Hello guys, I am new here to the forum and I was wondering if you could help me with some trouble I am having with differential equations. For your information: I am a second year applied physics student from the Netherlands and I loveee everything about physics and mathematics!
The problem concerns the Euler D.E.: x^2*y'' + x*y' + y = f(x). Let's assume for the moment that f(x) = 0, so that the equation is homogenous. I know how to solve it: by substituting y = x^r and calculating the two r's. I also know about the problem af the double root: the solution becomes y(x) = A*x^r1 + B*ln(x)*x^r1.
Well, this last 'solution' concerning the double root was kind of dropped during college: they just told us how to solve it. I am wondering where the ln(x) comes from. I have the same problem with linear D.E.'s with constant coefficients: y'' + y' + y = 0, if you substitute y=exp(kx) you get two k's unless there's a double root and then the solution becomes y(x) = A*exp(k1x) + B*x*exp(k1x). In this case I am wondering where the 'x' comes from. Can someone provide me with a proof for these solutions?
Thanks in advance!
EDIT: by proof I do not mean the proof that they ARE solutions. That's easy checkable: I just fill them in in the D.E.'s. I mean where it comes from, to help me understand it better.
The problem concerns the Euler D.E.: x^2*y'' + x*y' + y = f(x). Let's assume for the moment that f(x) = 0, so that the equation is homogenous. I know how to solve it: by substituting y = x^r and calculating the two r's. I also know about the problem af the double root: the solution becomes y(x) = A*x^r1 + B*ln(x)*x^r1.
Well, this last 'solution' concerning the double root was kind of dropped during college: they just told us how to solve it. I am wondering where the ln(x) comes from. I have the same problem with linear D.E.'s with constant coefficients: y'' + y' + y = 0, if you substitute y=exp(kx) you get two k's unless there's a double root and then the solution becomes y(x) = A*exp(k1x) + B*x*exp(k1x). In this case I am wondering where the 'x' comes from. Can someone provide me with a proof for these solutions?
Thanks in advance!
EDIT: by proof I do not mean the proof that they ARE solutions. That's easy checkable: I just fill them in in the D.E.'s. I mean where it comes from, to help me understand it better.