Given the differential equation: $$x^3y''−xy'+y=\frac{1}{x^2}e^{−\frac{1}{x}}, y(1)=y'(1)=\alpha$$ And the general solution of the differential equation to be: $$y(x)=c_1x+c_2xe^{\frac{−1}{x}}−\frac{e^{\frac{−1}{x}}}{3x^2}−\frac{e^{\frac{−1}{x}}}{x}−2e^{\frac{−1}{x}}.$$ What is the interval of definition for the IVP, where $\alpha$ is some positive real?
Is this the correct approach to find the interval of definition for an IVP?
Changing the differential equation to linear normal form: $$y''−\frac{1}{x^2}y'+\frac{1}{x^3}y=\frac{1}{x^5}e^{−\frac{1}{x}}$$ Now, looking at the coefficients of the y's and the forcing there are defined everywhere except x = 0. Also, given the initial condition is x = 1 $\implies$ the interval of definition is $(0, \infty)$