Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 4 / c

On either connected half-line, the homogeneous linear differential equation isIntegration gives $\boxed{y_h=C{x}^{-2}{e}^{-1/x}}$, including the zero solution. For the inhomogeneous equation choose the positive integrating factorIt satisfies ${\mu }^{\prime }/\mu =2/x-1/x^2$. Multiplication gives the exact derivativeHence all solutions on either half-line areFor $a\ne 0$, impose $y(a)=b$ to obtain $C=a^2{e}^{1/a}b-\arctan a$. The complete initial value problem solution is thereforeIt is smooth throughout the maximal connected interval $(0,\infty )$ if $a>0$, and $(-\infty ,0)$ if $a<0$.

For every $a\ne 0$ and every real $b$, there is exactly one solution on the maximal connected interval containing $a$. The formula in part (i) proves existence on the entire appropriate half-line, and the initial condition fixes its sole constant uniquely. This agrees with the local Picard-Lindelöf theorem.

For $a=0$, there is no classical solution on an interval containing the initial point, whatever $b$ is: the printed equation is not defined there. Even if one only imposes the equation for $x\ne 0$ and asks for a finite continuous extension across zero, the negative side rules it out. For $x\to 0^{-}$:

Thus no choice of the negative-side integration constant admits a finite left limit.

There is a useful distinction with a separately extended one-sided problem. On $x>0$, every member of the solution family tends to zero with all its derivatives as $x\downarrow 0$, since the exponential dominates every inverse power of $x$. If the forcing is separately defined to be zero at the right endpoint, that modified one-sided problem has infinitely many smooth solutions with $y(0)=0$, and none with $y(0)\ne 0$. Those one-sided extensions do not give solutions on the two-sided intervals in the question.