Confirm one solution and find general solution to homogenous equation

Question

Veryify that $y=e^x$ satisfies the homogeneous equation associated with $$(x-1)y''-xy'+y=1$$ and obtain the general solution.

I have tried the method of undetermined coefficients and that has not worked out.

How do I go about tackling this problem? Is there a method I am overlooking?

No asking for all the work, just some tips on where to start. Thanks so much.

Answer 1

I think you are intended to finish solving the homogeneous equation via reduction of order (especially since you are given one solution 'for free'), guess that $y = v(x)e^x$ is a solution, and then use the differential equation to formulate a new equation for $v$.

For finding the solution to the inhomogeneous problem, it sounds as though your method works, but you're simply overlooking the fact that $y$ appears on the left hand side, so in particular, if you guess $y_p = c$, then you simply have $y_p = 1$ after plugging it in to the equation.

I am not sure this was the intended method: this equation isn't actually suited for undetermined coefficients, because to write in the form $y'' + p(t)y' + q(t)y = g(t)$, we have $g(t) = \dfrac{1}{x-1}$, which is not amenable to the method. Nonetheless, the first and foremost way to solve differential equations is good guesswork, and simply verifying it is a solution is sufficient.

Answer 2

$(x-1)y''-xy'+y=1$

$(x-1)y''-xy'+y-1=0$

Let $u=y-1$ ,

Then $u'=y'$

$u''=y''$

$\therefore(x-1)u''-xu'+u=0$

$u=C_1x+C_2e^x$

$y-1=C_1x+C_2e^x$

$y=C_1x+C_2e^x+1$