 Solution

Write $y={\int }_C{e}^{xt}f(t)dt$ and move the factor $x$ onto $e^{xt}$ by integration by parts. The differential-equation residual isThus the Laplace integral solution is obtained fromup to a nonzero constant and branch choices. The endpoint factor is proportional to $t^b(1-t{)}^{a-b}$, so the given positive real parts ensure its vanishing at $0$ and $1$.

For $x>0$, choose the directed intervals $C_1=[0,1]$ and $C_2=(-\infty ,0]$; for $x<0$, choose $C_1=[0,1]$ and $C_2=[1,\infty )$. On each interval choose a continuous branch separately. Equivalently their integrands, after an irrelevant constant phase is removed, use $t^{b-1}(1-t{)}^{a-b-1}$ on $(0,1)$, $(-t{)}^{b-1}(1-t{)}^{a-b-1}$ on $t<0$, and $t^{b-1}(t-1{)}^{a-b-1}$ on $t>1$. Exponential decay at the infinite endpoint removes the remaining boundary term.

For large positive $x$, $C_1$ has a nonzero leading term proportional to $e^x{x}^{-(a-b)}$, while $C_2$ is proportional to $x^{-b}$. For large negative $x$, $C_1$ is proportional to $(-x{)}^{-b}$ and $C_2$ to $e^x(-x{)}^{-(a-b)}$. These distinct asymptotics prove independence in each case; the gamma factors are nonzero under the assumptions.

For $a=3,b=1$, integrate $e^{xt}(1-t)$ on $[0,1]$ to get $(e^x-1-x)/x^2$. A convenient independent pair on either side of zero isDirect substitution verifies both, and their ratio is not constant. Their singularities cancel in $y_2-y_1$. Since $e^x-1-x=x^2/2+O(x^3)$, the solution regular at zero with the required normalization isIts power series supplies the removable value and verifies the equation at the singular point.